s; 

*JSr 

lltivmitg 


Name  of  Book  and  Volume, 


Division 

Range 

Shelf..... 


GENERAL  THEOKT 


or 


BRIDGE  CONSTRUCTION: 


CONTAINING 


DEMONSTRATIONS  OF  THE  PRINCIPLES  OF  THE  ART 
AND  THEIR  APPLICATION  TO  PRACTICE ; 

FURNISHING    THE    MEANS    OF 

CALCULATING  THE  STRAINS  UPON  THE  CHORDS,  TIES,  BRACES, 

COUNTER-BRACES,  AND  OTHER  PARTS  OF  A  BRIDGE 

OR  FRAME  OF  ANY  DESCRIPTION. 

WITH  PRACTICAL  ILLUSTRATIONS. 


BY 

HERMAN  HAUPT,   A.M., 

CIVIL    ENGINEER. 


NEW   YOEK: 
D.    APPLETON    AND    COMPANY, 

549    &    551    BROADWAY. 
1873. 


**: 


ENTEB.ED,  according  to  Act  of  Congress,  in  the  year  1851,  \>j 
I).  APPLETON  &  COMPANY, 

to  the  Clerk's  Office  of  the  District  Court  for  the  Southern  District  of 
New -York. 


PREFACE. 


FKOM  the  great  importance  of  the  art  of  bridge  construction 
it  might  be  supposed  that  its  principles  would  be  familiarly 
understood  by  all  whose  occupations  require  an  acquaint 
ance  with  it.  If,  however,  any  work  exists,  containing  an 
exposition  of  a  theory  sufficient  to  account  generally  for 
the  various  phenomena  observed  in  the  mutual  action  of 
the  parts  of  trussed  combinations  of  wood  or  metals,  the 
author  has  neither  seen  or  heard  of  it.  "When  his  attention 
was  first  directed  to  the  subject  of  properly  proportioning 
the  parts  of  bridges,  by  being  called  upon  in  the  discharge 
of  professional  duties  to  superintend  their  construction,  he 
found  it  impossible  to  procure  satisfactory  information, 
either  from  engineers  and  builders,  or  from  books.  In  fact, 
if  he  may  be  permitted  to  judge  from  the  contradictory 

(5) 


6  PREFACE. 

opinions  that  are  even  yet  entertained,  and  from  the  errors 
frequently  committed  by  practical  men  in  the  construction 
of  bridges,  he  would  be  inclined  to  infer,  that  of  few  arts 
of  equal  practical  importance,  are  the  principles  so  little 
understood. 

The  best  works  on  the  subject  of  construction,  that  have 
fallen  into  the  hands  of  the  writer,  contain  but  little  that 
will  furnish  the  means  of  calculating  the  strains  upon  the 
timbers  of  a  bridge  truss,  or  of  determining  their  relative 
sizes;  and  they  do  not  furnish  information  as  to  what 
constitute  the  elements  of  a  framed  truss,  or  the  most 
advantageous  disposition  of  these  elements  to  attain  the 
maximum  strength  and  stiffness  with  a  given  quantity  of 
material. 

In  the  following  pages,  the  object  has  been,  not  so  much 
to  detail  particular  plans,  as  to  establish  general  principles. 

An  attempt  has  been  made  to  explain  the  mode  of 
action  of  the  parts  of  structures,  and  their  mutual  influence 
when  combined;  to  point  out  the  ways  by  which  the 
strains  can  be  estimated,  and  the  relative  sizes  of  the 
timbers  accurately  determined ;  new  combinations  of  the 
elements  in  the  construction  of  bridge  trusses  have  been 
suggested,  the  defects  of  many  plans  in  general  use  pointed 
out,  and  several  simple  means  proposed  for  remedying 
these  defects  and  adding  to  the  strength  of  structures. 

Tillable  to  procure  satisfactory  information  in  any  other 
way,  the  writer,  in  the  spring  of  1840,  commenced  a  course 
of  experiments  on  models,  and  examined  many  existing 
structures,  for  which  his  occupation  at  that  time  as  a  civil 


PREFACE.  7 

engineer  in  the  service  of  the  State  of  Pennsylvania,  afford 
ed  great  facilities. 

This  investigation  was  commenced  without  the  most  re 
mote  design  of  ever  giving  the  result  publicity,  but  solely 
for  the  purpose  of  enabling  him  to  proceed  intelligently  in 
the  discharge  of  professional  duties.  The  facts  elicited,  led 
him  to  conclude  that  many  important  structures,  and  some 
of  recent  erection,  exhibited  defects  which,  although 
serious,  had  escaped  the  observation  of  merely  practical 
builders ;  these  will  be  pointed  out  in  their  proper  places, 
and  the  writer  will  be  amply  rewarded  for  the  labor  of 
preparing  this  little  treatise,  if  it  shall  prove  the  means  of 
adding  in  the  smallest  degree  to  the  stock  of  information 
already  possessed  on  the  important  subject  of  bridge  con 
struction. 

The  treatise  on  bridge  construction  has  been  prefaced 
by  a  few  pages  on  the  resistance  of  solids,  because  a 
knowledge  of  this  subject  lies  at  the  foundation  of  the  art 
of  construction.  The  mode  of  investigation  is  peculiar, 
and  is  believed  to  be  more  simple  than  those  usually 
employed. 

This  part  of  the  subject  will  not  be  interesting  to  those 
who  are  unacquainted  with  the  application  of  mathematics 
to  mechanics,  but  those  who  are,  will  perhaps  be  pleased 
with  the  simplicity  of  the  solutions,  and  the  novelty  of 
representing  strains  by  geometrical  solids,  deflections  by 
parabolic  areas,  and  the  variable  pressures  at  different, 
points  of  beams  by  the  corresponding  ordinates  of  plane 
curves.  The  portion  which  refers  to  the  principles  of 


8  PREFACE. 

/ 

construction,  and  their  application  to  practice,  will  be 
readily  understood  by  the  general  reader.  Demonstrations 
of  propositions  required  in  the  solution  of  problems,  and 
not  found  in  any  work  accessible  to  the  author,  have  been 
given  in  notes. 

It  did  not  form  part  of  the  original  design  to  include  the 
subject  of  the  construction  of  stone  arches,  but  it  was 
thought  that  the  work  would  be  rendered  more  useful, 
complete,  and  generally  acceptable,  if  a  few  pages  were 
added  containing  a  simple  exposition  of  the  principles 
of  this  important  art. 

Mathematicians  have  apparently  exhausted  their  inge 
nuity  in  devising  modes  of  distributing  the  weights  so  as 
to  produce  an  equilibrated  curve  of  suitable  form  for  the 
intrados  of  an  arch ;  but  many  of  their  speculations  are  far 
more  curious  than  useful,  whilst  practical  men  have  been 
disposed  to  reject  the  principle  of  equilibration  as  inappli 
cable  to  constructions.  It  was  a  long  time  before  the 
fortunate  discovery  was  made  that  the  intrados  might  be 
of  any  form  most  pleasing  to  the  eye,  and  that  the  con 
ditions  of  equilibrium  could,  in  general,  be  satisfied  by 
making  the  joints  of  the  voussoirs  perpendicular  to  the 
line  of  direction  of  the  pressures;  a  fact  so  simple  and 
obvious,  that  there  is  reason  for  surprise  that  it  was  not 
suggested  to  the  first  mind  in  which  originated  the  idea  of 
an  arch  of  equilibrium.  This  principle  is  important  in  its 
practical  results,  and  an  admirable  application  is  made  of 
it,  by  John  Seaward,  a  British  Engineer,  in  a  work  con 
taining  a  proposed  plan  for  the  London  Bridge.  This 


PREFACE.  9 

gentleman,  however,  treats  his  subject  in  such  a  way  that 
only  an  expert  mathematician  would  attempt  to  follow 
him ;  his  book  is  consequently  of  little  value  to  the  practi 
cal  builder,  and  the  methods  usually  given  for  determining 
experimentally  the  curve  of  pressure,  are  of  very  difficult 
application,  and  the  results  of  doubtful  accuracy.  The 
formulas  for  determining  the  thickness  of  abutments,  as 
'given  by  Hutton  and  others,  are  based  upon  principles 
which  are  not  quite  practically  correct,  and  give  results  too 
small  to  be  relied  upon. 

Influenced  by  these  considerations,  the  writer  has  been 
induced  to  propose  a  method  for  determining  the  curve 
of  equilibrium,  or  in  other  words  the  line  of  direction  of 
the  pressures,  which  he  believes  to  be  new,  simple,  and 
easy  of  application. 

If  this  volume  shall  be  found  to  possess  no  other  merit, 
the  author  can  at  least  claim  that  it  is  not  a  compilation. 
All  the  prepositions,  with  perhaps  one  or  two  exceptions, 
have  been  proved  by  modes  of  demonstration  which  he 
believes  to  be  entirely  new,  and  different  from  those 
employed  by  Tredgold,  Gregory,  Hutton,  and  other  writers 
on  mathematics  or  mechanics :  he  was  so  situated  at  the 
time  when  his  attention  was  first  directed  to  these  subjects, 
that  he  could  not  procure  the  works  of  the  standard  writers 
above  referred  to  for  reference,  and  was  therefore  led  to 
originate  principles,  and  modes  of  demonstration,  which 
subsequent  comparison  causes  him  to  think  are  different 
from  any  heretofore  employed,  and  in  most  instances, 
simple  and  direct.  The  principle  already  referred  to  of 


10  PREFACE. 

determining  the  strains  of  beams  by  the  volumes  of  geo 
metrical  solids,  and  the  deflection  and  extension  of  the 
fibres  by  a  comparison  of  the  areas  of  plane  curves,  appears 
not  to  have  been  previously  employed ;  at  least  the  writer 
has  never  met  with  any  thing  of  the  kind,  as  far  as  his 
acquaintance  with  the  writings  of  mathematicians  extends. 
In  the  hope,  that  results  which  have  proved  of  great 
value  to  him  may  not  be  entirely  useless  to  others,  and  in 
the  belief  that  the  theory  which  ho  has  advanced  will 
explain  the  phenomena  observed  in  the  mutual  action  of 
the  parts  of  bridges,  and  furnish  the  means  of  proportion 
ing  them  upon  correct  principles ;  the  author  submits  the 
result  of  his  labors  to  the  consideration  of  those  who  are 
interested  in  the  theory  or  practice  of  bridge  construction. 


CONTENTS 


PART  I. 


RESISTANCE  OF  MATLRIALS  ...............................  .......  19 

Flexure  ..................................................  22 

Beams  supported  at  both  ends  ......................  •>  .......   24 

Inclined  beams  ..........................................  -  •  26 

Strength  of  short  rectangular  posts  ..........................  26 

Flexure  of  columns  and  posts  ........................  *  ......  32 

Resistance  of  posts  to  flexure  ...............................   34 

Tension  ..................................................   38 

Torsion  .........................  ,  ........................   39 

Forms  of  equal  strength  ........................  .  ..........  41 

Influence  of  the  vertical  forces  ..............................   43 

Relative  deflections  .......................................  48 

Strength  of  particular  sections  ..............................  50 

Means  of  determining  the  constants  ..........................  59 

WOODEN  BRIDGES  .............................................  63 

Horizontal  strain  .........................................   65 

Vertical  force  at  any  point  ..................................    65 

To  find  the  curve  which  represents  the  horizontal  strain  ........   71 

To  find  the  pressure  upon  the  supports  when  a  beam  is  framed  as 
a  cap  upon  the  tops  of  several  vertical  posts,  and  a  weight 
applied  directly  over  one  of  the  posts  ....................  73 

Strength  of  a  long  beam  laid  over  several  supports  .............  76 

Effects  of  counter-bracing  ................  .  .................   82 

(ID 


12  CONTENTS. 

Inclination  of  braces 84 

To  determine  the  strain  upon  counter-braces 86 

To  determine  the  strain  upon  braces  and  ties 87 

Of  the  strain  upon  the  ties  and  braces  at  the  centre 89 

Effects  upon  the  braces  and  ties  which  result  from  the  introduc 
tion  of  arch-braces 01 

To  determine  the  strains  upon  the  chords 94 

Means  of  increasing  the  strength  of  bridge  trusses 101 

On  the  maximum  span  of  a  wooden  bridge 104 

Effects  of  counter-bracing  upon  an  arch 10G 

Roadway 108 

CAST-IRON  BRIDGES 110 

APPLICATION  OF  RESULTS 114 

To  determine  the  strain  upon  the  chords 115 

Of  the  strain  upon  the  lower  chord,  at  the  centre 110 

Strain  at  the  ends  of  the  chords 11G 

Of  the  strain  upon  the  ties  and  braces 117 

Counter-braces 120 

Lateral  horizontal  braces 120 

Diagonal  braces , 121 

Floor  beams 121 

Amount  of  counter-bracing  which  an  arch  requires 123 

EQUILIBRIUM  OF  ARCHES 125 

To  find  the  thickness  of  abutments  of  arches  of  any  kind 125 

To  find  the  relative  length  of  the  joints  at  different  points  of  an 

arch,  and  the  line  of  direction  of  the  pressure 130 

ILLUSTRATIONS  OF  PARTICULAR  MODES  OF  CONSTRUCTION 14-1 

Foot  bridge  across  the  river  Clyde 141 

Bridge  over  the  torrent  of  Cismore 142 

Bridge  across  the  Portsmouth  river 144 

Timber-bridge  over  the  river  Don,  at  Dyce  in  Aberdeenshire. . .  145 

Schaffhausen  bridge 145 

Long's  bridge 146 

LATTICE  BRIDGES 148 

IMPROVED  LATTICE 152 

Columbia  bridge 155 


PART  H. 

PREFACE 161 

PENNSYLVANIA  RAILROAD  VIADUCT 107 

Superstructure 169 


CONTENTS.  18 

Bills  of  materials  for  one  span 171 

Bill  of  castings 172 

Bill  of  bolts 172 

Arch  suspension  bolts 172 

Weight  of  nuts  for  one  span , 173 

Estimate  of  cost  do 173 

Workmanship 173 

Principles  of  calculation 174 

Calculation  of  the  strength  of  the  bridge  on  the  supposition  that 

the  arch  sustains  the  whole  weight 175 

Strain  upon  the  arch  suspension  rods ' 176 

Strain  upon  the  counter-bracing  produced  by  the  action  of  the 

arch 17C 

Strength  of  the  truss  itself  without  the  arch 178 

Strain  upon  the  ties 179 

Strain  upon  the  floor  beams , . . .  182 

Strain  upon  the  counter-braces 183 

Lateral  braces 184 

Strain  upon  the  diagonal  braces 185 

Resistance  to  sliding  upon  the  supports 187 

Power  of  resistance,  on  the  supposition  that  the  arches  and  truss 

form  but  a  single  system 188 

Estimate  of  the  longitudinal  strains 189 

General  summary 192 

Strains  upon  the  parts  when  both  systems  are  united 193 

COYE-RUN  VIADUCT 194 

Description 194 

Bill  of  materials  for  one  span 195 

Bill  of  malleable  iron 195 

Wood 19G 

Recapitulation 196 

Calculation 196 

Estimate  ef  cost 197 

IRON  BRIDGE  ACROSS  HARFORD-RUN,  BALTIMORE 197 

Description 197 

Bill  of  materials 199 

Wood  for  the  whole  bridge 20C 

Recapitulation  of  bill  of  materials 200 

Estimate  of  cost 200 

Estimate  of  cost  of  a  single  track  bridge,  with  two  trusses 201 

Calculation. . , 201 

Principle  of  calculation , 202 

Recapitulation 203 

LITTLE  JUXIATA  BRIDGE 203 

Description 202 


14  CONTENTS. 

Bill  of  materials  for  one  span 20tj 

Malleable  iron  for  one  span 207 

Nuts 207 

Summary 208 

Estimate  of  cost  of  one  span. 208 

Workmanship , 208 

Data  for  calculation. , 209 

Calculation  of  strains 209 

Strain  upon  the  chords 210 

Strain  upon  the  posts 211 

Strain  upon  the  ties 211 

Lateral  and  diagonal  braces 212 

Floor  beams 212 

Counter-braces 213 

Strain  upon  the  counter-braces 214 

Vertical  pressure  upon  the  arch  and  posts 217 

General  summary  of  results 217 

SHERMAN'S  CREEK  BRIDGE,  PENN.  CENTRAL  RAILROAD 219 

Bill  of  timber  for  one  span 220 

Bill  of  counter-brace  rods,  for  one  span 221 

Arch  suspension  rods,  for  one  span 221 

Floor  beam  rods,  for  one  span 222 

Small  bolts,  for  one  span 222 

Dimension  and  data  for  calculation. 222 

Calculation  of  truss  without  the  arches 223 

Ties  and  braces 223 

Floor  beams 225 

Lateral  braces 225 

Strain  upon  the  knee-braces 226 

Pressure  upon  the  arch 228 

Vertical  pressure 230 

Estimate  of  cost  of  one  span 233 

Summary 233 

RIDER'S  PATENT  IRON  BRIDGE 234 

Description 234 

Bill  of  materials  for  a  single  span,  60  ft 234 

Cast-iron 234 

Malleable  iron 234 

Wood 235 

Approximate  estimate  of  cost 235 

Calculation  235 

CUMBERLAND  VALLEY  RAILROAD  BRIDGE 237 

Bill  of  timber  for  one  span,  186  ft 238 

Iron  rods . . . , , 239 

Estimate  of  cost  of  one  span 239 


CONTENTS.  15 

Data  for  calculation 235 

Strain  upon  the  ties   24C 

I'RENTON  BRIDGE 242 

IRON  ARCHED  BRIDGE,  of  133  feet  span 243 

Description 243 

IRON  BRIDGE  OVER  RACOON  CREEK 247 

Bill  of  materials 248 

Estimate 251 

Workmanship 251 

Data  for  calculation 252 

BALTIMORE  AND  OHIO  RRILROAD  BRIDGE 253 

Description  of  details 253 

CANAL  BRIDGE,  PENN.  RAILROAD 254 

BOILER  PLATE  TUBULAR  BRIDGE 255 

ARCHED  TRUSS  BRIDGE,  READING  RAILROAD 257 

BRIDGE  ACROSS  THE  SUSQUEHANNA 258 

Bill  of  timber 259 

IMPROVED  LATTICE  BRIDGES. 260 

TRUSSED  GIRDER  BRIDGES 265 


PART  I. 


RESISTANCE  OF  TIMBER 


AND 


OTHER  MATERIALS 


CALCULATIONS  for  the  purpose  of  determining  the  relations 
which  the  dimensions  of  timbers  should  bear  to  the  weights 
which  they  are  required  to  sustain,  are  based  upon  several  hy 
potheses  'which  experience  has  proved  to  be  correct  within  the 
usual  practical  limits. 

The  most  important  of  these  are — 

1.  The  fibres  are  susceptible  of  compression  and  extension. 

2.  The  degree  of  extension  or  compression  will  be  directly 
as  the  force  by  which  it  is  produced. 

3.  So  long  as  the  elasticity  remains  unimpaired,  or  so  long 
as  the  fibres  may  be  considered  as  perfectly  elastic,  the  force 
required  to  produce  a  given  extension  will  be  equal  to  that 
which  produces  an  equal  compression,  and  the  resistances  to 
these  forces  will  be  likewise  equal. 

These  hypotheses  will  be  applied  to  the  most  simple  case 
of  flexure,  which  is,  that  of  determining  the  relations  between 
an  applied  weight  and  the  dimensions  of  a  timber  which  are 
necessary  to  sustain  it  when  one  end  is  fixed  and  the  other 
unsupported. 

(19} 


20  BRIDGE   CONSTRUCTION. 

FIG.  1. 


"WO 

Let  A  0  represent  a  beam  fixed  at  A  and  loaded  at  0  with 
a  weight  (w\  the  weight  of  the  beam  itself  being  for  the  present 
disregarded — 

The  substance  of  the  beam  is  supposed  to  be  entirely  uni 
form  throughout,  and  composed  of  an  assemblage  of  parallel 
fibres,  all  being  equally  strong. 

The  effect  of  the  weight  w  is  to  stretch  the  fibres  at  A  and 
compress  those  at  B.  From  these  points  to  the  interior  of  the 
beam  the  forces  gradually  dimmish,  and  there  must  exist  some 
point  of  the  line  A  J5,  at  which  no  horizontal  force  is  exerted,  and 
at  which  the  fibres  suffer  neither  extension  or  compression. 

To  that  line  of  the  longitudinal  section  which  passes  through 
this  point,  parallel  to  the  direction  of  the  beam  A  (7,  has  been 
given  the  name  of  the  neutral  axis,  a  term  which  will  hereafter 
be  very  frequently  employed. 

The  position  of  the  neutral  axis  will  vary  with  the  form  of 
the  beam,  with  the  degree  of  uniformity  which  it  possesses,  and 
with  the  amount  of  flexure  caused  by  the  load ;  but  in  a  beam 
that  is  straight-grained,  rectangular,  'without  knots  or  flaws  of 
any  kind,  a'nd  not  subjected  to  the  action  of  a  weight  sufficient 
to  impair  its  elasticity,  it  is  practically  correct  to  assume  the 
position  of  the  neutral  axis  in  the  middle  of  the  section. 

Admitting,  then,  that  within  the  usual  practical  limits,  it  is 
sufficiently  correct  to  assume  the  position  of  the  neutral  axis  in 
the  middle  of  the  beam,  it  is  evident  that  from  this  line  in  the 
direction  n  A  and  n  B  the  pressures  on  the  fibres  will  increase 
directly  as  the  distance,  and  if  the  pressure  upon  any  fibre  at 

the  distance  —be  designated  by  Jft,  the  pressure  upon  any  other 

fibre  may  be  determined  from  a  simple  proportion.  The  total 
pressure  upon  the  line  n  B  can  then  be  directly  determined ;  for 
since  the  pressure  upon  any  individual  fibre  is  as  the  distance 


RESISTANCE   OF   MATERIALS.  21 

from  the  neutral  axis,  it  would  be  represented  by  the  perpen 
dicular  erected  upon  the  base  (J  d)  of  a  right  angled  triangle 
whose  altitude  is  72,  and  the  whole  pressure  would  be  represen 

7?        /7  7? 

ted  by  the  area  of  this  triangle  or  by  ^  d  X  —  =  -~ — 

FIG.  2. 


B —  — R — 

The  several  forces  which  act  upon  the  beam  may  be  con 
sidered  as  tending  either  to  cause,  or  to  prevent  motion  around 
the  point  n,  and  their  effects  must  be  ascertained  by  comparing 
the  products  of  their  intensities  by  the  distances  from  the  point 
of  rotation  at  which  they  act. 

If,  for  example,  a  weight  should  be  applied  at  the  extremity 
of  a  lever,  its  effect  would  not  be  represented  by  the  weight 
alone,  but  by  the  weight  multiplied  by  the  distance  from  the 
fulcrum  at  which  it  acts ;  this  product  is  the  moment  of  the 
force,  and  it  is  these  moments,  in  reference  to  the  axis  or  point 
of  rotation,  and  not  simply  the  absolute  intensities  of  the  forces, 
that  must  be  compared  in  determining  the  conditions  of  equi 
librium  in  any  system. 

Now  the  weight  of  any  body  may  be  supposed  concentrated 
at  its  centre  of  gravity ;  and,  in  general,  any  number  of  parallel 
forces  may  be  replaced  by  a  single  force  called  the  resultant. 
In  the  present  case,  the  pressure  of  the  triangle,  which  repre 
sents  the  sum  of  all  the  forces  upon  the  fibres  of  the  lower  half 
of  the  section  A  B,  will  be  the  same,  as  if  a  single  force  equal  to 
its  area  was  applied  in  the  direction  of  a  line  passing  through 
its  centre  of  gravity. 

As  the  centre  of  gravity,  or  centre  of  parallel  forces  of  a  tri 
angle,  is  in  a  line  drawn  from  the  vertex  to  the  middle  of  the 
base,  and  at  a  distance  from  the  latter  equal  to  one-third  the 
length  of  the  bisecting  line,  it  follows  that  the  leverage  of  the 
triangle  of  pressure  will  be  two-thirds  of  n  B  or  J  d ;  this  mul 
tiplied  by  the  area  of  the  triangle,  (i.  e.),  by  the  resisting  force 

along  n  B  which  we  have  found  to  be  equal  to  -j-,  will  give 


22  BRIDGE   CONSTRUCTION. 

for  the  moment  of  this  force,  in  reference  to  the  point  n,  — j-  X 

d  _    R  d2 
3" =      12    ' 

But  the  part  n  A  opposes  a  resistance  to  extension  which  is 
equal  to  that  which  the  part  n  B  opposes  to  compression,  and, 
as  the  moments  of  these  forces  are  equal,  the  whole  moment  of 

the  resisting  forces  will  be  expressed  by  — - — . 

The  weight,  which  is  represented  by  w,  acts  with  a  leverage 
equal  to  the  length  of  the  beam,  and  its  moment  will  therefore 
be  (w  T) : 

The  equation  of  equilibrium  will  therefore  be  Wl  =  — ^ — . 

In  this  equation  the  breadth  of  the  beam  has  been  regarded 
as  unity,  but  if  it  be  represented  by  (b)  the  equation  will  become 


The  value  of  R  must  be  determined  by  experiment  and 
will  depend  upon  the  kind  of  material.  In  general,  it  has  been 
taken  too  high,  and,  as  a  consequence,  the  dimensions  of  timbers 
deduced  from  the  formula  which  contained  it  have  been  too 
small. 

Timber  should  never  be  subjected  to  a  strain  sufficient  to 
destroy  its  elasticity,  and  experiments  to  determine  the  value  of 
M  should  be  continued  for  a  considerable  length  of  time. 

A  weight  which  even  after  several  months  would  produce 
any  permanent  flexure  should  be  regarded  as  too  great. 

"When  a  beam  is  used  as  part  of  a  frame,  such  a  value  must 
be  given  to  the  constant  in  the  proper  formula,  that  only  a  very 
slight  degree  of  flexure  will  take  place :  the  limit,  assigned  by 
Tredgold,  being  one-fortieth  of  an  inch  for  every  foot  in  length, 
or  4 J<j  I 

Flexure. 

To  determine  the  conditions  of  flexure,  let  it  be  supposed 
that  a  beam  is  fixed  at  one  end  and  loaded  at  the  other, 
the  material  being  perfectly  elastic.  It  is  evident,  in  the 


RESISTANCE   OF   MATERIALS.  23 

first  place,  that  the  deflection  will  be  directly  as  the  weight, 
(i.  e.),  if  a  given  weight  produces  a  given  deflection,  twice  that 
weight  will  produce  twice  that  deflection.  Again,  the  deflection 
will  be  proportional  to  the  length  and  the  strain  upon  the  fibres, 
the  last  of  which  is  represented  by  J?,  and  as  R  contains  I  in 
the  numerator  it  will  consequently,  for  these  reasons  alone  con 
sidered,  be  as  the  square  of  the  length ;  but,  again,  the  deflection, 
or  which  is  the  same  thing,  the  depression  at  the  extremity,  will 
be  directly  proportional  to  the  amount  by  which  the  fibres  are 
extended  or  compressed,  which  is  also  as  the  length,  and,  there 
fore,  the  deflection  must  be  as  the  length  cubed. 

To  make  this  subject  more  clear,  let  w  represent  the  weight 
suspended  at  the  extremity  of  a  beam  whose  length  is  Z,  and 
let  us  ascertain  the  deflection  when  I  becomes  2  I. 


Let  Apr  p"  represent  a  horizontal  line,  and  suppose  that  the 
action  of  the  weight  causes  a  deflection  equal  to  pf  n,  it  is  ob 
vious  that  by  increasing  the  length  to  nf,  other  considerations 
omitted,  the  deflection  would  \>Q  p"  nf  —  2  pf  n. 

But  if  the  point  of  application  of  w  be  transferred  from  n  to 
B,  the  leverage  will  be  doubled,  R  will  be  doubled,  and  the  de 
flection  again  doubled  from  this  cause. 

Again,  the  deflection  will  be  as  the  length  of  fibre  extended. 

For  example,  if  a  given  weight  should  extend  a  rod  ^  of  an 
inch,  the  same  weight  would  extend  a  rod  of  twice  the  length  -^ 
or  %  of  an  inch,  and  as  the  deflection  must  be  proportional  to 
this  extension,  it  follows  that  the  deflection  must  be  again 
doubled  from  this  third  cause ;  hence,  the  deflection  will  be 
directly  as  the  cube  of  the  length. 

6  w  I 
Lastly,  as  the  deflection  is  as  the  value  of  72,  which  is    ,    ,2  y 

it  will  be  inversely  as  the  breadth  and  the  square  of  the  depth  : 


24  BRIDGE   CONSTRUCTION. 

but  the  quantity  of  angular  motion  around  the  point  of  support 
to  which  the  deflection  is  proportioned  is  also  inversely  as  the 
depth,  as  may  be  seen  by  reference  to  the  figure ;  in  which,  if 
A  n  becomes  A  n'  =  2  A  n,  the  deflection  will  become  m  mf  — 
•J  B  m,  since,  if  the  leverage  of  the  resistance  be  doubled,  the 
effect  will  be  reduced  one-half.  Hence,  it  follows  that  the  de 
flection  will  be  inversely  as  the  cube  of  the  depth. 

FIG.  4. 


Combining  all  these  results,  it  follows  that  the  whole  deflec 
tion  will  be  directly  as  the  weight  and  the  cube  of  the  length, 
and  inversely  as  the  breadth  and  the  cube  of  the  depth — and 

W  I3 
will  be  expressed  by    ,   ,3  . 

If  different  timbers  be  required  to  fulfil  the  condition,  that 
the  deflection  shall  be  equal  whatever  be  the  length,  we  have 
only  to  make  this  expression  constant,  and  determine  its  value 
by  direct  experiment  upon  the  particular  kind  of  timber  to  be 
used. 

The  condition  of  equal  stiffness,  however,  does  not  require 
that  the  deflection  should  be  equal  for  every  length,  but  allows 
it  to  be  in  proportion  to  the  length :  for  example,  a  beam  of  20 
feet  may  be  allowed  to  bend  twice  as  much  as  one  of  10  feet, 

w  I2 
and  the  expression  modified  to  suit  this  case  will  be    ,    ,3  ,  a 

constant  quantity  for  beams  of  equal  stiffness. 

By  introducing  a  suitable  number  for  the  constant,  the 
equation  which  expresses  its  value  will  determine  any  one 
of  the  four  quantities,  w,  ?,  5,  or  c?,  when  the  other  three  are 
known. 

Beams  supported  at  both  ends. 

When  a  beam  rests  on  two  points  of  support,  and  is  loaded 
with  a  weight  applied  in  the  middle,  the  general  circumstances 
of  the  case  are  involved  in  that  which  we  have  considered. 


RESISTANCE   OF   MATERIALS.  25 

FIG.   5. 


If  w  represent  the  weight  applied  at  the  centre  0,  this  weight 
will  be  transmitted  equally  to  the  two  points  of  support  at  A 
and  B,  and  the  beam  may  therefore  be  considered  as  subjected 
to  the  action  of  two  forces,  each  (J  w)  acting  with  the  leverage 
(J  T)  against  a  fulcrum  0. 

As  the  expression  for  the  moment  of  the  resistance  is,  as 

R  d'z 
formerly,  —  g  —  ,  and  that  of  the  weight  J  w  X  J  I,  the  equation 

Ed2        wl  3wl 

of  equilibrium  becomes  —  p  —  =  ~^j~>  whence  R  =  Q  1,72  >  tne 

expression  which  gives  the  relative  dimensions  when  a  beam  is 
supported  at  both  ends. 

If  flexure  be  regarded,  the  same  equation  that  was  obtained 
in  the  case  of  beams  fixed  at  one  extremity  will  be  applicable  ; 
the  only  change  required  is  in  the  value  of  the  constant,  for 
which,  see  table  at  the  end  of  this  treatise. 

When  the  weight  is  not  in  the  centre. 

In  this  case,  let  c  represent  the  distance  of  the  point  of  ap 
plication  Qr  from  the  centre,  then  the  proportion  of  the  weight 
sustained  by  B  will  be  determined  from  the  proportion. 

w 
I  :  (}  I  +  c)  :  :  w  :  j  (  J  I  +  c). 

The  portion  sustained  by  A  will  be 


The  moments  of  these  forces,  in  reference  to  the  point 
will  be 


j  (JZ  +  c}  X  (II—  c)  =  T-( 


Z  +  c}  X  (II—  c)  =  T-(T—  °2}  =  moment  of  the  force  at  5. 


y(H  —  c)  x  (i  I  +  c)  =  T(T—  °2)  =  moment  of  the  force  at  B. 
These  moments  are  equal  to  each  other,  and  as  the  resistance 
of  the  beam  is  denoted  by  —  —  ,  the  equation  of  equilibrium 


26  BRIDGE   CONSTRUCTION. 

Eld2       w(P  —  ±  c2) 
will  be  — g-  —^ -', 

Qw(p 4C2\ 

whence,  R  —  — ,  ,    ,2  , — -  when  the  load  is  not  in  the  middle. 

Inclined  Beams. 

The  formulae  for  horizontal  timbers  are  equally  applicable 
to  those  which  are  inclined  by  taking  the  horizontal  distance 
between  the  extreme  points,  or  the  cosine  of  the  inclination  for 
the  value  of  I. 


Let  the  weight  w,  suspended  at  (7,  be  resolved  into  the  two 
components  On  and  n  o ;  the  latter,  being  parallel  to  the  fibres, 
will  be  destroyed  by  the  resistance  of  the  fixed  point  A.  The 
other,  perpendicular  to  the  fibres,  is  the  only  one  to  be  con 
sidered. 

The  value  of  this  force  is  expressed  by  Co  X  cos.  o  On  = 
w  cos.  B  A  P,  and,  substituting  this  value,  in  the  formula,  R  — 

3wl  Slcos.BAP 

0  ,   72,  it  becomes  R  = ?TT~T2 >  which  differs  from  the 

2  b  d2'  2  b  d2 

former  only  in  containing  I  X  cos.  B  A  P,  or  A  P  in  place  of  I. 

Strength  of  Short  Rectangular  Posts. 

The  investigation  of  this  subject  appears  to  present  consid 
erable  difficulty,  arising  from  the  fact,  that  the  point  of  applica 
tion  and  the  direction  of  the  pressure  are  often  indeterminate, 
and  if  an  attempt  is  made  to  express  by  a  formula  the  conditions 
of  equilibrium,  it  is  necessary  to  assume  data  which  may  only 
approximate  to  the  probable  condition  of  things  in  practice. 
Fortunately,  the  difficulty  is  more  theoretical  than  practical ;  if 


RESISTANCE   OF   MATERIALS.  27 

a  post  be  long  and  composed  of  elastic  materials,  the  condition, 
that  it  shall  be  of  sufficient  dimensions  to  resist  flexure  with  a 
given  applied  weight,  brings  the  question  into  a  tangible  form, 
and  the  solution  is  simple ;  on  the  other  hand,  if  the  post  is 
short,  the  usual  limit  of  the  weight,  which  is  generally  one 
tenth  of  that  which  would  be  required  to  crush  the  material,  is 
amply  sufficient  to  compensate  for  any  variation  in  the  point  of 
application,  or  line  of  direction  of  the  pressure  that  may  be  pro 
duced  by  unequal  settling  or  other  causes. 

If  we  attempt  to  continue  the  subject,  as  heretofore,  by 
establishing  relations  between  the  moments  of  the  acting  and 
resisting  forces,  the  first  step  must  be  the  determination  of  the 
position  of  the  neutral  axis ;  as  no  comparison  of  moments  can 
be  made  until  we  know  the  point  or  axis  of  rotation  to  which 
they  are  referred. 

In  posts  placed  vertically  and  loaded  at  their  upper  extrem 
ities  the  position  of  the  neutral  axis  can  no  longer  be  assumed 
in  the  centre,  but  will  vary  greatly  according  to  the  amount  and 
point  of  application  of  the  weight,  and  the  degree  of  flexure  that 
is  supposed  to  have  been  produced. 

It  appears  reasonable  to  conclude,  that  when  a  post  is  uni 
form  in  composition  and  acted  upon  by  a  force  applied  exactly 
in  the  direction  of  the  axis,  all  parts  of  the  central  cross  section 
are  subjected  to  nearly  equal  degrees  of  pressure ;  but  if  the 
weight  be  applied  at  either  side  of  the  axis,  the  compression  no 
longer  continues  uniform  but  becomes  greatest  on  that  side 
towards  which  the  pressure  is  applied. 

FIG.  7. 


-  ---------  n 


Let  A  B  CD  represent  the  longitudinal  section  of  a  rectan 
gular  post  of  some  stiff  material :  when  the  weight  is  applied 
at  w  every  part  of  the  cross  section  (n  n')  may  be  considered  as 


28  BRIDGE   CONSTRUCTION. 

subjected  to  equal  pressure,  and  if  R  and  R'  represent  the  forces 
acting  upon  the  fibres  at  n  and  nf,  these  forces  may  be  represen 
ted  by  some  portion  of  their  lines  of  direction,  as  np"  and  nr  p. 

In  the  present  case,  these  forces  are  supposed  equal,  and  the 
line  pp"  will  be  parallel  to  nnfm,  the  point  of  intersectioii3 
which  determines  the  distance  of  the  neutral  axis,  will  therefore 
f  be  infinite. 

If  w  be  applied  at  Wf  the  pressure  at  nr  would  become  great 
er  than  at  n,  and  if  npf  and  nf  p  represent  the  relative  magni 
tudes  of  the  forces  at  n  and  n',  the  line  p pf  will  be  inclined  to 
n  n'  and  must  intersect  at  some  point  0,  and,  consequently,  the 
distance  of  the  neutral  axis  will  become  finite  and  could  be  de 
termined  if  we  knew  the  relative  values  of  the  forces  at  n  and  n' . 

As  w  approaches  B,  the  differences  of  the  forces  at  n  and  nf, 
which,  for  brevity,  will  be  called  R  and  R',  will  become  greater, 
and  0  will  approach  n. 

After  the  post  yields  laterally,  the  fibres  along  A  C  will  be 
extended,  and,  before  it  arrives  at  this  point,  there  must  be  a 
certain  magnitude  of  the  weight,  or  a  certain  position  of  its  line 
of  application,  that  will  cause  neither  extension  or  compression 
along  A  (7,  which  will  accordingly  become  the  neutral  axis. 

With  a  still  greater  weight,  the  fibres  along  A  Q  being  ex 
tended,  and  those  along  B  D  compressed,  the  neutral  axis  must 
be  within  the  post  at  some  point  0' . 

If  the  post  is  long,  and  the  pressure  be  supposed  still  to  in 
crease,  the  elasticity  of  the  timber  being  unimpaired,  Of  will 
approach  very  near  the  centre.  Lastly,  a  still  greater  increase 
of  weight,  and  consequent  flexure,  will  destroy  the  elasticity, 
and  the  position  of  the  neutral  axis  will  then  depend  on  the 
relative  powers  of  the  fibres  to  resist  the  crushing  or  extending 
forces. 

Let  it  be  assumed,  that  the  direction  of  the  weight  coincides 
with  the  edge  of  the  post,  and  that  A  0  suffers  neither  exten 
sion  or  compression,  the  neutral  axis  will  be  at  n,  and  R,  as 
formerly,  representing  the  maximum  force,  which  will  be  at  %', 

7? 
the  resistance  will  be  expressed  by  (nnf  —  1)  X  -^,  its  moment 

will  be  — 5—  .  |  b  =  • — Q — •     The  moment  of  the  weight  being 


RESISTANCE   OF   MATERIALS. 


(b  w\  the  equation  of  equilibrium  becomes  — '-^ 
IR 


29 
=  6  w,  or W 


FIG.  8. 


When  the  weight  is  applied  directly  in  the  axis,  every  part 
of  the  section  n  nf  sustains  an  equal  portion.*  \Ye  have  there 
fore  w  =  b  R.  Hence  it  appears,  that  a  post  will  sustain  three 
times  as  much,  when  the  weight  is  applied  along  the  axis,  as  it 
will  when  the  line  of  direction  coincides  with  one  of  the  sides, 
provided,  the  dimensions  are  such,  that  flexure  can  take  place 
only  in  the  direction  of  b. 

Tredgold,  in  his  treatise  on  cast  iron,  page  234,  makes  the 
resistance  one-fourth  when  the  weight  acts  on  one  edge  of  a 
block.  This  requires,  that  the  neutral  axis  should  be  within 
the  rectangle  at  a  distance  from  the  line  of  pressure  equal  to  f 
the  breadth,  and  that,  beyond  the  axis,  the  parts  oppose  no  re 
sistance,  a  supposition,  which,  we  think,  is  less  nearly  correct 
than  that  which  we  have  assumed. 

*  This  would  be  true  in  all  cases  were  the  material  perfectly  unelastic  ; 
but  if  the  lateral  cohesion  of  the  fibres  be  not  so  great  as  to  prevent  any 
motion,  some  variation  in  the  degree  of  pressure  upon  different  parts  of 
the  cross  section  must  ensue.  This  will  certainly  be  the  case  when  the 
support  is  very  short  in  proportion  to  its  width  :  for  example,  a  weight 
applied  at  A  would  produce  a  tendency  to  flexure  in  the  direction  of  the 
dotted  lines,  and  then  the  pressure  &tp  would  be  greater  than  at  n  or  n'. 
This  objection  is  of  no  practical  importance,  as  supports  are  always  too 
long  to  allow  of  flexure  in  this  way. 


FIG.  9. 

A 


30 


BRIDGE   CONSTRUCTION. 


We  will  continue  the  hypothesis,  that  the  neutral  axis  is  on 
one  side  and  the  direction  of  the  pressure  on  the  other. 

When  the  line  of  direction  of  the  weight  coincides  with  the 
axis  of  a  column,  the  strength  ivill  be  '/>  as  great  as  when  it 
coincides  with  one  side. 

FIG.  10. 
c         A 


Let  AB  represent  the  line  of  direction  of  the  weight,  02) 
=  the  neutral  axis,  R  =  strain  upon  n'9  the  strain  upon  any 
point  p  would  be  represented  by  a  perpendicular  through  that 
point  terminated  by  the  oblique  plane  A  n,  the  whole  pressure 
would  consequently  be  represented  by  the  semi-cylinder  A  n  nr. 
The  vertical  line,  through  the  centre  of  gravity,  passes  at  a  dis 
tance  from  n  =  f  radius.* 

*  As  the  centre  of  gravity  of  this  solid  is  not  given  in  any  mathemati 
cal  work  to  which  the  author  has  access,  he  thinks  it  proper  to  explain 

5 
the  method  by  which  he  has  obtained  the  distance  -r  r. 

To  find  the  volume  and  centre  of  gravity  of  a  semi-cylinder  cut  off  by 
an  oblique  plane  passing  through  the  edge  of  the  base. 

FIG.  11. 


Let  r  =  radius,  x  —  any  abscissa,  y  the  corresponding  ordinate  of  the 
circle. 

T> 

Then  2  r :  x : :  R :  -5 —  x  =  perpendicular  of  elementary  rectangle. 


RESISTANCE   OF   MATERIALS.  31 


The  moment  will  therefore  be  — ^ —  •  R  •  f  r  =  %  if  r3  R. 

The  moment  of  the  weight,  acting  with  a  leverage  2  r,  is  2 
w  r.     The  equation  of  equilibrium  is  f  *  r3  R  =  2  w  r,  or  T6C  •»• 

-75 —  x'2y  =  area  =  —  xy.    —  xydx  =  elementary  solid. 

T> 

—  yx*dx  =  moment  of  elementary  rectangle. 

But   from    the    equation    of  the    circle    we   havey=\/2ra;  —  xa 

*R  R  i 

—  x  y  d  x  =  — f  (2r  x  —  x2)?  x  d  x     make  (r  —  x}  =  z,  d  x  =  — 

r> 

dz,  2  r  x  —  x2  =  r2  —  z2.     Substitute  these  values  we  obtain  —  /(2  rx  — 


/(r2— z*)$zdz].  The  first  of  these  integrals  taken  between  the  limits  -f  r 
and — ris  the  area  of  a  semi-circle  and  is  consequently  equal  to  — ^ —  hence 

rt  r3 
the  value  of  the  first  term  becomes  — ~ — . 

The  second  term  becomes ^ "  (see  demonstration  of  ungula,  Prob. 

14),  and  is  equal  to  0  when  z  =  -f  r,  or  z  =  —  r;  it  therefore  disappears, 
and  the  volume  of  the  solid  becomes  —  —  •  — „ —  =  —  |-  it  r2  R ;  a  result 
which  is  evidently  correct,  since  the  volume  is  half  that  of  the  cylinder  rt  r  2E. 

7-> 

To  find  the  distance  to  the  centre  of  gravity  we  must  divide  the  integral  of — 
yx2dx  by  the  volume.  Making  similar  substitutions  to  those  used  in  finding 
the  volume,  we  obtain  —  fy  x2dx  =  —f  (r2  —  z2}  %  (r  —  z}  *  (—  d  z)  = 


i 

The  first  of  these  integrals  is  a  semi-circle,  hence  r*f(r2  —  z2}  ?  d  z  =  r* 
2  x  ra  *r* 

2  ~2~' 

The  second, /(r2  —  z2}"2 zdz,  as  we  have  seen,  becomes  =  0. 

The  third,/(r2  —  z2}^  z2  d  z,  is  proved,  in  the  problem  of  the  ungula,  to 
be  Jr*/(r«  —  z2}^dz,  which,  between  the  limits  4-  r  and  — r,  becomes 

«r4 

-— .     (See  note  to  Prob.  14.) 


32 


BRIDGE   CONSTRUCTION. 


r2  R  =  iv.  But  when  the  pressure  coincides  with  the  axis,  we 
have  cr  r2 R  —  w\  hence,  the  strength  in  the  two  cases  will  be 
as  5  to  16. 

Flexure  of  Columns  and  Posts. 

It  is  evident  that  if  a  column  be  perfectly  cylindrical,  and 
the  direction  of  the  weight  coincide  exactly  with  the  axis,  flex 
ure  cannot  take  place ;  but  if  the  weight  be  sufficient  the  fibres 
will  yield  by  crushing.  Flexure  therefore  must  result  from 
some  obliquity  in  the  line  of  direction  of  the  force. 

FIG.  12. 


Let  us  take  the  most  unfavorable  case  that  would  probably 
occur  in  practice,  as  it  is  that  which  gives  the  greatest  diameter 
and  is  consequently  the  most  safe. 

Let  A  be  the  point  of  application  of  the  weight,  A  B  its  line 
of  direction,  CD  the  position  of  the  neutral  axis  at  the  instant 


T> 

The  whole  expression  therefore  becomes  — fy  x2dx 
It 


R  {  XT' 
~~  r  V    2 


Hence,  the  line  through  the  centre  of  gravity,  perpendicular  to  the  "base, 
passes  at  a  distance  of  £  r  from  the  centre  ;  the  centre  of  gravity  will  be 
found  in  this  line,  and  also  in  the  line  drawn  from  A  to  the  middle  point 
of  B  C;  hence,  it  will  be  at  their  intersection,  and  its  height  above  the 

base  can  be  found  by  the  proportion  2  r  :  —  :  :  |  r  :  -^  R. 


RESISTANCE    OF    MATERIALS.  33 

of  flexure.     The  force  at  A,  in  the  direction  A  B,  acts  with  a 
leverage  n  n',  and  tends  to  produce  rotation  around  the  point  n. 

Join  A  U,  and  let  the  weight  (w)  be  represented  by  the  por 
tion  A  P  of  its  line  of  direction.  By  constructing  the  parallelo 
gram  of  forces  on  A  n  and  A  n',  it  will  be  evident  that,  in  con 
sequence  of  the  obliquity  of  the  line  A  n,  there  will  result  a 
horizontal  component  (A  0'),  the  magnitude  of  which  will  b 
proportional  to  the  cosine  of  the  inclination  A  n  nf.  If  A  be 
fixed,  as  it  always  is  in  columns,  so  that  it  cannot  move  in  the 
direction  A  </,  the  force  A  o  will  be  transmitted  to  n,  and  its 
horizontal  component,  =  A  </,  will  produce  a  cross  strain  upon 
the  middle  of  the  column  at  n. 

The  reaction  of  the  point  B,  upon  which  alone  the  column  is 
supposed  to  rest,  produces  a  force  =  w,  and  acting  in  an  opposite 
direction,  its  horizontal  component  at  n  will  be  equal  to  A  0', 
and  the  fibres  at  n  will  be  subjected  to  a  cross  strain  resulting 
from  the  actions  of  these  two  forces  equal  to  2  A  o'. 

The  vertical  component  is  resisted  by  the  strength  of  the 
material,  and,  as  it  is  the  horizontal  force  alone  which  tends  to 
produce  flexure,  this  alone  will  be  considered. 

As  the  column  is  fixed  at  the  points  A  and  By  and  subjected 
to  a  cross  strain  at  n,  it  is  in  the  condition  of  a  beam  supported 
at  both  ends  and  loaded  in  the  middle,  and,  therefore,  the  con 
clusions  at  which  wre  arrive  in  the  former  case  are,  with  slight 
modification,  applicable  here. 

It  was  shown,  that  beams  to  be  equally  stiff  must  be  pro- 

w  I2 
portional  to  -rjjy  but  in  the  case  of  a  cylinder  b  =  d,  and 

wl2 

the  expression  becomes  ~TT,  in  which  d  represents  the  di 
ameter. 

The  expression  for  the  strain  \~jr)  shows  that  it  is  direct 
ly  as  the  weight  and  square  of  the  length,  and,  inversely,  a& 
the  fourth  power  of  the  diameter ;  *  but  the  weight  does  not,  in 

*  That  the  strain  is  inversely  as  the  fourth  power  of  the  diameter  may 
also  be  shown  by  the  following  considerations  : 

3 


34  BRIDGE    CONSTRUCTION. 

this  case,  represent  the  pressure  on  the  top  of  the  column,  but 
the  cross  strain  at  the  middle. 

The  value  of  the  constant,  in  rectangular  beams,  was  deter 
mined  by  the  condition,  that  the  flexure  should  be  ^  of  the 
length,  or  ^  of  an  inch  per  foot.  The  same  constant  would 
give  for  a  column  10  feet  high  a  deflection  of  J  of  an  inch.  If 
this  be  considered  too  great,  the  constant  must  be  increased : 
but  it  must  be  remembered  that  this  is  the  maximum  deflection, 
on  the  supposition,  that  the  weight  is  thrown  altogether  upon 
one  side  of  the  column,  the  most  unfavorable  case  that  can 
occur ;  it  is  therefore  probable  that  no  change  in  the  value  of 
the  constant  is  required. 

When  the  height  of  a  column  does  not  exceed  about  nine 
times  the  diameter,  it  is  found,  that  the  fibres  will  crush  before 
they  will  yield  laterally,  and  the  strength  will  therefore  be  pro 
portional  to  the  area  of  the  section,  or  d2 ;  we  have  in  this  case, 


Resistance  of  Posts  to  Flexure. 
The  ordinary  formula  for  the  stiffness  of  beams,  supported 


If  A  D  be  supposed  to  represent  the  neutral  axis,  and  R  the  maximum 
strain  upon  the  fibres  B  C;  the  pressure  upon  any  part  of  the  section 
n  nf  would  be  represented  by  a  perpendicular  to  n  n'  terminated  by  the 
oblique  plane  p  n.  The  solid  n  n' p,  whose  altitude  R  is  constant,  and 
whose  base  is  equal  to  the  area  of  the  section,  will  therefore  represent 
the  pressure  upon  n  n',  and  will  be  proportional  to  d2.  The  leverage, 
being  the  distance  from  n  to  the  perpendicular,  through  the  centre  of 
gravity,  will  also  be  proportional  to  d,  and  therefore  the  strength  of  the 
cross  section  would  be  in  proportion  to  d3,  and  the  strain  inversely  as  ds. 
The  strain  will  also  be  as  the  deflection,  which,  as  in  the  case  of  hori 
zontal  beams,  can  be  shown  to  be  inversely  as  the  diameter ;  hence,  com 
bining  all  these  results,  the  strain  will  be  inversely  as  d  . 


RESISTANCE    OF    MATERIALS.  35 

at  the  ends  and  loaded  in  the  middle,  is 

_bd3 
c  12>> 

in  which  (w)  represents  the  weight  which  produces  a  given 
deflection,  b  =  breadth  in  inches,  d  =  depth  in  inches,  and  I  = 
length  in  feet ;  c  is  a  constant,  to  be  determined  by  substituting 
the  values  of  the  other  quantities  in  the  equation. 

In  making  experiments  to  determine  the  constant  from  this 
formula,  it  is  necessary  to  observe  very  accurately,  both  the 
weights  and  the  deflections  produced  by  them,  and  then,  by 
means  of  a  proportion,  find  the  value  of  (w),  which  will  pro 
duce  the  deflection  required  to  be  substituted  in  the  formula. 

In  reflecting  upon  the  circumstances  connected  with  the 
flexure  of  beams,  the  writer  conceived  the  idea  of  deducing  an 
expression  for  the  weight  which  a  post  would  support  from  the 
ordinary  formula  for  the  stiffness  of  a  horizontal  beam,  by  the 
following  considerations.  If  a  beam  is  bent  by  an  applied 
weight,  there  will  be  a  tendency,  from  the  elasticity  of  the 
material,  to  recover  its  form  when  the  weight  is  removed ;  but 
if  the  ends  are  fastened  by  being  placed  between  resisting 
points,  so  that  the  piece  cannot  recover  its  shape,  there  must 
be  a  horizontal  force  caused  by  the  reaction  of  the  material ; 
and  this  force  is  such,  that  if  the  beam  were  placed  in  a  verti 
cal  position  and  loaded  with  a  weight  equal  to  it,  the  deflection 
should  be  the  same  as  that  of  the  horizontal  beam,  and  conse 
quently  the  extreme  limit  of  the  resistance  of  the  post  to  flexure 
would  be  determined. 

To  ascertain  the  force  which  is  exerted  by  the  reaction  of  a 
bent  beam  in  the  direction  of  the  chord  of  the  arc. 

FIG.  13. 


c 

Let  A  B  represent  a  beam,  supported  at  the  ends  and  load 
ed  with  a  weight  (w)  applied  at  the  middle  point. 
d  =  deflection  caused  by  the  applied  weight. 
B  0  =  tangent  of  curve  at  B. 


36  BRIDGE    CONSTRUCTION. 

If  the  weight  be  removed,  the  reaction  of  the  beam  will 
cause  it  to  regain  its  original  figure  if  not  resisted  by  a  pressure 
at  the  ends.  The  force  of  this  reaction  will  be  proportional  to 
the  degree  to  which  the  fibres  are  strained,  and  as  the  strain 
upon  the  fibres  is  nothing  at  the  ends  A  and  JB,  and  increases 
uniformly  to  the  middle  point,  the  force  of  reaction  will  be  in 
the  same  proportion,  and  the  point  of  application  of  the  result 
ant  of  the  whole  of  the  reacting  forces  will  correspond  to  the 
centre  of  gravity  of  a  triangle,  whose  base  is  Bf;  it  will  con 
sequently  be  at  a  distance  from  B  =  f  B  f. 

The  effect  of  this  resultant  acting  at  a  distance  f  B  /,  must 

be  the  same  as  the  weight  (—  \  acting  at  a  distance  J5/,  and 

€fl 

must  consequently  be  in  proportion  to  —  as  3:2.      The  va 

lue  of  the  resultant  is  therefore  —  —  . 

4 

The  line  of  direction  of  the  pressure  at  B  being  the  tangent 
B  (7,  the  force  of  reaction  at  h  may  be  considered  as  applied  at 
the  point  Jc  of  its  line  of  direction,  and  as  kliB  and  OfB 
are  similar  triangles,  Of:  fB  ::  |  w  :  horizontal  pressure 


at  B  =  |  w  X     —=  I  wjj-;  =    -.Q-,  -     Representing  this 
force  by  P  we  have 

P  =     ^wl 

~Wd" 

As  the  deflection  of  a  b<;am  within  the  elastic  limits  is 
always  in  proportion  to  the  weight,  if  (wf)  =  the  weight  that 
will  produce  a  deflection  equal  to  unity,  the  deflection  (d)  will 
require  a  weight  =  (d  w'\  and  by  substituting  this  value  in  the 
equation,  we  find 

n        9       dwf 

P  =  -I\'  —J 

In  this  expression  (d\  which  represents  the  deflection,  has 
disappeared,  and  as  (wf)  is  a  constant  quantity  for  the  same 
beam,  representing  the  weight  that  produces  a  deflection  equal 


RESISTANCE   OF   MATERIALS.  37 

to  the  unit  ot  measure,  it  follows,  that  P  is  the  same  with  every 
weight  and  every  degree  of  deflection  within  the  elastic  limits. 

This  result  seems,  at  first  view,  to  be  contrary  to  fact  ;  it 
would  appear  that  if  the  weight  is  increased,  the  horizontal 
strain  should  be  increased  in  the  same  proportion  ;  but  when  it 
is  remembered,  that  the  deflection  increases  with  the  weight, 
and  that  the  former  diminishes  the  value  of  P  in  precisely  the 
same  proportion  that  the  latter  increases  it,  the  difficulty  van 
ishes,  and  the  reason  why  P  should  be  constant  for  the  same 
beam  becomes  obvious. 

The  practical  importance  of  this  result  is  very  great,  as  it 
furnishes  the  means  of  obtaining  a  formula,  which  will  give  at 
once  the  extreme  limit  of  the  resistance  to  flexure,  or  the  weight 
which,  applied  to  a  post,  will  cause  it  to  yield  by  bending. 

As  the  formulae  used  by  Tredgold  are  calculated  for  a  de 
flection  of  4\>  of  an  inch  to  one  foot,  or  ?J^  of  the  length,  the 
weight  which  would  cause  a  deflection  of  1  would  be  w 

/  I  v       480  w 

{  1  -f-  TO/T)  =  —  T  —  ?  and  by  substituting  this  value  for  wf  in 

the  equation 

P  =  &t*'J, 

we  find  P  =  90  w  =  A. 

But  from  the  ordinary  formula  for  the  stiffness  of  a  beam 
supported  at  the  ends,  we  have 


bd3  90  b  d 


w  =  —  TO.     Therefore  P  =  —  TO—  =  B. 

c  I2  clz 

The  expression  P  =  90  w  shows  that  the  extreme  limit  of 
the  strength  of  any  post  whatever,  of  any  length,  breadth,  or 
depth,  or  of  any  kind  of  material,  is  ninety  times  the  weight 
which  causes  a  deflection  of  7|o  °f  tne  length. 

90  b  d3 
The  second  expression,  P  =  -  —  j^~  ->  w^  g*ve  tne  value  of 

0  if 

P  directly,  without  first  knowing  the  weight  required  to  cause 
*  given  deflection  in  a  horizontally  supported  beam.  In  this 
expression,  b  =  breadth  in  inches,  d  =  depth  or  least  dimension 
in  inches,  I  =  length  in  feet,  and  c  =  a  constant  to  be  deter 
mined  by  experiment  for  each  species  of  material. 

The  value  of  c  for  white  pine  is  -01.     By  substituting  this 


38  BRIDGE   CONSTRUCTION". 


9000 


value,  we  find  P  = =-2 ,  a  remarkably  simple  formula, 

L 

which  gives  the  extreme  limit  of  the  resistance  to  flexure  of  a 
white  pine  post. 

The  same  expression  may  be  employed  to  determine  the 
constants  used  in  the  ordinary  formula  for  the  stiffness  of 

90  bd3 

beams.    For  this  purpose  let  the  equation  P  = r^—  be  trans- 

c  I 

90  b  d3 
posed,  which  will  give  c  =     -p  ,a  .     Find  P  by  applying  a 

string  to  a  flexible  strip  of  the  material  to  be  experimented 
upon,  in  the  manner  of  a  chord  to  an  arc,  and  ascertain  the 
tension  on  the  chord  with  an  accurate  spring  balance.  It  will 
be  found  that,  whether  the  strip  be  bent  much  or  little,  the 
tension  on  the  chord,  as  shown  by  the  spring  balance,  will  be 
constant,  and  this  tension,  in  pounds  substituted  for  P,  will 
give  the  value  of  c  without  requiring,  as  is  necessary  with  other 
formulae,  an  observation  of  the  deflection. 

Experiments  made  upon  these  principles  with  strips  of 
white  pine,  yellow  pine,  and  white  oak,  5  feet  long,  1|-  inches 
wide,  and  \  inch  deep,  gave  the  following  results :  — 

The  observed  tensions  were, 

White  Pine,  7J  Ibs.  value  of  c  =  '0097 
Yellow  Pine,  6J  "  "  "  =  -0108 
White  Oak,  6|  "  "  "  ==  -0104 

As  the  stiffness  is  inversely  as  these  constants,  it  follows 
toat  white  pine  is  stiffer  than  yellow  pine  or  oak.  The  experi 
ments  of  Tredgold  give  similar  results. 

Tension. 

When  a  force  is  applied  in  the  direction  of  the  axis  of  a 
suspension  rod,  the  resistance  is  directly  proportional  to  the 
area  of  the  section ;  and,  consequently,  it  is  only  necessary  to 
multiply  this  area  by  the  number  expressing  the  resistance  of 
a  square  inch.  As  metals  are  the  only  substance  well  suited 
to  resist  tensile  strains,  we  find  that  they  are  almost  exclusive 
ly  employed  for  this  purpose,  and  generally  in  such  lengths, 


RESISTANCE   OF   MATERIALS.  39 

when  compared  with  their  diameters,  that  the  directions  of 
the  strains  will  always  coincide  very  nearly  with  the  axis. 

If,  however,  in  extreme  cases,  it  should  happen,  that  the 
weight  can  be  thrown  altogether  on  one  side,  then  it  is  neces 
sary  to  increase  the  area  of  the  section  ;  the  amount  of  which 
increase  will  be  determined  upon  similar  principles  to  those 
which  apply  in  the  case  of  columns. 

Torsion. 

The  resistance  which  is  opposed  by  a  shaft  to  a  twisting 
force  is  called  resistance  to  torsion. 

The  following  method  of  investigation  is  similar  to  that 
pursued  by  Tredgold. 

If  the  shaft  be  of  the  form  of  a  rectangular  plate,  we  may 
suppose  it  to  be  supported  at  the  corners  A  and  B,  and  weights 
suspended  from  each  of  the  other  corners  0  and  D  ;  the  strain 
produced  by  loading  it  in  this  manner  would  be  similar  to  the 
twisting  strains  in  shafts.  As  the  weights  at  the  four  corners 
are  supposed  equal,  there  would  be  an  equal  tendency  to  break 
at  the  same  time  along  the  diagonals  A  B  and  (7.Z),  or  along 
some  other  lines,  at  which  the  resistance  might  be  less ;  but, 
before  fracture,  one  of  these  lines  will  serve  as  a  fulcrum  for 
forces  acting  at  the  extremity  of  the  other. 


To  determine  the  line  of  fracture  when  the  material  is  uni 
form  in  composition  and  equally  tenacious  in  every  direction, 
let  z  represent  the  length  A  B  of  the  line  of  least  resistance,  b 
=  the  breadth  of  the  plate  B  0.  Then  A0=  V  z2—b2,  and  if 
V  represent  the  perpendicular  Op,  we  have  z:  V z2 — b2::b 


itO  BP.IDGE    CONSTRUCTION. 

The  weight  acting  at  the  point  (7,  its  effect  will  be  in  pro 
portion  to  the  leverage  Op,  and  the  resistance  of  the  section, 
the  depth  being  constant,  will  be  as  its  length  z.  The  line  of 
least  resistance  is  that  at  which  z  will  be  least  when  compared 

ij 

with   11.    or    when  —  is    a   maximum.      We    have    therefore 
*'  z 


^  z  2 £  2 

— I — ,  which  is  a  maximum  when  z  =  b  </  2,  or  A  C  =  B  0. 

The  case  which  has  been  considered  is  the  most  simple,  and 
applies  to  the  resistance  of  a  uniform  material  such  as  cast  iron. 

Having  found  the  length  of  2,  it  is  only  necessary  to  sub 
stitute  its  value  for  the  breadth,  in  the  ordinary  formulae  for  the 
resistance  to  a  cross  strain,  in  order  to  determine  the  strength. 

This  calculation  is  evidently  founded  on  the  supposition, 
that  the  resistance  is  the  same  in  every  direction,  as  is  the  case 
in  cast  iron  and  similar  materials  ;  but,  in  shafts  composed  of  a 
fibrous  substance,  the  line  of  least  resistance  would  have  a  dif 
ferent  direction,  depending  upon  the  relative  proportion  between 
:he  lateral  and  longitudinal  cohesions.  It  is  not  necessary, 
however,  that  we  should  proceed  to  the  examination  of  this 
case,  since  the  only  questions  of  practical  importance  are : 

What  is  the  resistance  to  flexure  within  the  elastic  limits, 
and  what  degree  of  angular  motion  will  be  produced  by  a 
given  flexure  ? 

The  angle  of  torsion  may  be  deduced  from  the  following 
considerations.  Let  (e)  be  the  extension  which  the  material  will 
bear  without  injury  when  its  length  is  unity.  This  extension 
must  obviously  limit  the  movement  of  torsion,  or  the  angle  of 
torsion  ;  but,  since  the  line  of  greatest  strain  in  a  bar  of  great 
er  length  than  the  diagonal  of  the  square  of  its  base  is  in  the 
direction  of  the  diagonal  of  a  square,  the  whole  extension 
would  be  in  proportion  to  the  length  of  a  line  wrapped  around 
the  bar  at  an  angle  of  45°  with  the  axis,  and  would  therefore 
be  equal  to  (leV2).  The  arc  described,  or  the  angular  motion 
which  this  extension  will  allow,  is  equal  to  the  diagonal  of  a 
square,  of  which  this  extension  is  the  sides,  or  1  :  •jZule 
\/2  :  2  I  e,  as  is  obvious  by  reference  to  the  figure ;  the  exten- 


RESISTANCE   OF   MATERIALS.  41 

eion  of  the  diagonal  =  w,  admitting  a  degree  of  angular  motion 
=  m. 

FIG.  15. 


2  le  represents  the  arc  described  in  feet,  or  24  I  e  =  the  arc 
in  inches.     But  if  a  =  the  number  of  degrees  in  an  arc,  and 

•5-  its  radius,  -0174533  being  the  length  of  an  arc  of  one  de- 

CL  d 
gree  when  its  radius  is  unity,  we  have  24  I  e  —  -^-  X  .0174533, 

2750  I  e 

or  a  = ^ — . 

a 

That  is,  the  angle  of  torsion  (a)  is  as  the  length  and  extensi 
bility  of  the  body  directly,  and  inversely  as  its  diameter. 

2-284  I 
The  value  of  e  for  cast  iron  is  j^,  hence  a  —  -  — ^ — . 

The  value  of  e  for  malleable  iron  is  7^33,  hence  a  =  — -, — . 


Forms  of  equal  strength  for  beams  to  resist  cross  strains. 

Whatever  may  be  the  form  of  the  beam,  it  is  always 
necessary  that  the  area  of  the  section  resting  upon  the 
points  of  support  should  be  sufficient  to  resist  the  force  of 
detrusion,  or  that  which  tends  to  crush  the  fibres  in  a  direction 
perpendicular  to  their  length.  This  resistance  is  directly  pro 
portioned  to  the  area,  and  if  w  represent  the  weight  at  the  point 
of  support,  R  the  resistance  per  square  inch,  I'  =  breadth,  and 
df  the  depth ;  then,  the  dimensions  at  the  end  must  never  be 
less  than  will  be  given  by  the  equation  w  =  R  bf  df. 

The  practice  of  other  writers  has  been  to  omit  the  conside 
ration  of  this  force  in  determining  the  forms  of  equal  strength. 


42  BRIDGE    CONSTRUCTION. 

The  following  results,  obtained  by  omitting  for  the  present 
the  consideration  of  the  detrusive  force,  agree  with  theirs. 

PROPOSITION  1.  If  a  learn  be  supported  at  the  ends,  and 
the  load  applied  at  any  point  between  the  supports,  the  ex 
tended  side  being  straight,  the  form  of  the  compressed  side  will 
be  that  formed  by  two  semi-parabolas. 

Fia.  16. 
c 


Consider  0,  the  point  of  application  of  the  weight,  as  & 
fulcrum. 

Let  w  represent  the  portion  of  the  weight  sustained  by  A. 

x  =  the  distance  to  any  section  whose  depth  is  y. 

The  strain  will  be  w  x,  the  resistance  proportional  to  the 
square  of  the  depth  will  be  y2 ;  hence  y2  =  ID  x,  which  is  the 
equation  of  a  parabola. 

PROP.  2.     If  the  depth  be  constant,  the  horizontal  section 

d2 
will  be  a  trapezium.    For  in  this  case  iv  x  =  d2  y,  x  -=•  —  y :  or 

y  is  proportional  to  x,  which  is  the  property  of  a  triangle. 

FIG.  IT. 

PROP.  3.  Wlien  a  beam  is  regularly  diminished  towards 
the  points  that  are  least  strained,  so  that  all  the  sections  ivill 
be  circles,  or  other  similar  figures,  the  outline  should  be  a  cubic 
parabola. 

For,  in  this  case,  if  y  represent  the  diameter,  or  side  of  the 
section,  the  resistance  will  be  as  y*,  and  the  equation  of  mo 
ments  will  be  w  x  =  y 3. 

The  same  figure  is  proper  for  a  beam  fixed  at  one  end  and 
the  force  acting  at  the  other. 


RESISTANCE    OF    MATERIALS.  43 

PROP.  4.  If  a  weight  be  uniformly  distributed  over  the 
length  of  a  beam  supported  at  both  ends,  and  the  breadth  be 
the  same  throughout,  the  line  bounding  the  compressed  side 
should  be  a  semi-ellipse  when  the  lower  side  is  straight. 

FIG.  18. 


A--- 


ji' --~m'~ — -i  0" 

The  reaction  of  the  points  of  support,  A  and  B,  may  be  con 
sidered  as  two  forces  acting  upwards,  whilst  the  uniform  weight 
acts  downward  ;  these  two  classes  of  forces  are  therefore  op 
posed  to  each  other,  and  the  strain  at  any  point  will  be  proper- 
tional  to  the  difference  of  their  moments. 

Call  I  —  -\  A  B.         iv  =  weight  on  A. 

The  moment  of  w  at  the  distance  x  will  be  w  x. 

?/)  jT 

The  portion  of  the  weight  on  x  will  be  — — -,  its    moment 

c 


10  X        X  W  X 2 


I       2         2T 

Hence,  wx =  y 2      .*.  w  (2  Ix  —  x2)  =  2  I y*. 

—  I 

To  refer  the  curve  to  the  centre  (7,  make  I  —  x  =  z,  whence 
21  x  —  x2  =  I2  —  z2,  and  by  substitution  we  have  w  I2  —  wx2. 
=  2ly2,or2ly2  +  wx2  —  wl2,  which  is  the  form  of  the  equa 
tion  of  an  ellipse  referred  to  the  centre. 


Influence  of  the  vertical  forces. 

The  forms  which  are  here  given  for  beams  of  equal  strength 
correspond  with  the  results  obtained  by  all  who  have  written  up 
on  the  subject  of  the  strength  of  materials.  But  that  they  are  not 
strictly  correct  can  be  readily  proved.  They  have  been  obtained 
by  directing  attention  only  to  the  horizontal  forces  which  pro 
duce  longitudinal  strains  upon  the  fibres  of  the  beam.  But 
another  force  (the  existence  and  effects  of  which  will  be  more  ful 
ly  considered  when  we  treat  of  wooden  bridges),  appears  to  have 


44  BRIDGE   CONSTRUCTION. 

been  disregarded.  Dr.  Young  alludes  to  a  force  which  he  calla 
detrusion,  the  effect  of  which  is  to  crush  across  the  fibres  close 
to  a  fixed  point,  but  no  allusion  is  made,  either  by  him  or  any 
other  writer,  (as  far  as  our  information  extends,)  to  the  exist 
ence  of  a  force  acting  transversely  on  the  fibres  at  any  other 
point.  It  will  be  shown  hereafter,  that  when  the  beam  is  uni 
formly  loaded,  the  vertical  force  in  the  centre  is  nothing,  and  it 
increases  uniformly  to  the  ends,  where  it  is  equal  to  half  the 
weight  upon  the  beam.  Consequently,  if  the  breadth  of  a  beam 
is  constant  (Fig.  18),  the  true  figure  of  equal  strength  will  not 
be  A  s  n  but  o  m  n,  in  which  the  area  of  B  o  must  be  sufficient 
to  resist  detrusion  at  the  point  B,  and  o  m  c  must  be  a  straight 
line. 

If  the  beam  was  a  solid  of  revolution,  a  s  n  or  n  m  B  would 
be  a  cubic  parabola,  and  c  m  o  a  common  parabola. 

If,  instead  of  being  uniformly  distributed,  the  weight  were 
applied  entirely  at  the  centre,  the  form  of  equal  strength  would 
be  determined  by  the  intersection  of  n'  m'  B  with  the  side  of 
the  rectangle  Co',  and  would  be  n'  m"  of. 

Lastly,  if  there  should  be  both  a  uniform  load  and  a  weight 
applied  at  the  middle,  the  figure  to  resist  the  vertical  forces 
would  be  a  triangle  placed  upon  a  rectangle  (a  trapezoid),  and 
the  form  of  equal  strength  in  this  case  would  be  n'  m'  o" . 

PROP.  5.  If  a  beam  uniformly  loaded  and  depth  constant, 
be  supported  at  the  ends,  the  outline  of  the  breadth  should  be 
two  parabolas. 

FIG.  19. 


*7/  J  /7* ""' 

The  strain  upon  the  section  y  is  w  x — -.     The  resist 
ance,  since  the  depth  is  constant,  will  be  proportional  to  y. 

Hence,  (w  x — ^  =  y. 

Substitute  I — z  for  ar,  to  refer  to  the  centre  09  it  becomes 


RESISTANCE  OF    MATERIALS. 


Make  z  —  o,  we  have  for  the  ordinate  n  c  =  y  — 


45 


wl2 


To  refer  the  curve  to  the  point  n,  we  must  make  yr  =  n  e  — 

iv  I2                         w(l2 — z2}      wl2        , 
y,ory  =  »tf— y  =  — y';  hence,         g/        =  TJTJ Jr- 

Aft 

Reducing,  we  have  yr  =  -~-y  £2>  which  is  the  equation  of  a 
parabola. 

PROP.  6.  If  a  beam  is  fixed  at  one  end  only,  the  breadth 
constant,  and  the  weight  uniformly  distributed,  the  form  of 
equal  strength  will  be  a  triangle. 

FIG.  20. 


The  weight  on  x  is ,  the  moment  is     9      ,  the   moment 

of  resistance  ?y2. 


TT  o       w  x2  /  w  A,  f 

Hence,  y  =  ,  or  y  —  \ /  — -•  x,  the  equation  of  a 

L  /  »       Zi  I 

straight  line. 

If  the  weight  be  all  at  one  end  we  have  y2  =  w  x,  a  para 
bola. 

If  the  sections  be  similar  figures  y^  =  w  x,  a  cubic  para 
bola. 

PROP.  7.  The  form  of  a  suspension  rod  of  equal  strength 
is  determined  by  the  equation  #  — 2  a  log.  y. 

FIG.  21. 


46  BRIDGE    CONSTRUCTION. 

Let  w  =  weight  suspended  at  A 
P  =  weight  of  any  portion  of  the  rod  A  o 
x  =  A  o,  y  =  o  m,  dx  =  oof,y-{-dy  =  of  n 
a  =  resistance  of  material  per  square  inch. 
Then,  (w  +  P)  =  ay2 
y2  d  x  =  solidity  or  weight  of  portion  o  o' 
w  +  P  -f  y2  d  x  =  a  (y  +  d  y}2  (subtracting  the  first  equa 
tion)  y2dx=-2aydy 

j         o       dy  0     , 

d  x  =  2  a  — —         x  =  2  a  log.  y. 

y 

PROP.  8.  The  strength  of  two  similar  cylinders,  or  other 
solids  of  the  same  material,  supported  at  the  ends  and  strained 
ly  their  own  weights,  will  be  inversely  as  their  like  dimen 
sions. 

The  strength  of  a  single  cylinder  would  be  proportional  to 

d3 

Let  another  similar  cylinder  be  n  times  the  dimension 

w  I 

of  the  first,  its  weight  would  be  as  1  :  n3,  and  its  strength 

(n  d3}  d3 

— i '- —  = r ;  hence,  the  strength  of  the  first  would  be 

n  I  x  n   w       n w  I 

to  the  strength  of  the  second  as  n  :  1.  If  the  span  of  a 
bridge  be  doubled,  even  if  the  dimensions  of  all  the  parts  be 
increased  in  the  same  proportion,  the  strength  will  only  be 
one-half. 

PROP.  9.  The  parabolic  beam  of  equal  strength  contains 
two-fifths  less  material  than  the  circumscribing  cylinder. 

FIG.  22. 


\y 


From  the  equation  of  the  curve  we  have  y2  =  x.      Hence, 
the  element  of  the  surface  A  B  0  '=  x  dy  =  y3  dy,  and  the 


. 

The  volume  is  equal  to  the  surface  multiplied  by  the  cir 
cumference  described  by  the  centre  of  gravity. 


RESISTANCE   OF   MATERIALS. 


47 


f    fj      =  ^stance  of  centre  of 
2  ff  •  |  y  •  i  x  y  —  §  *  y    x  —  I  (cylinder  generated  by 
revolution  of  .A  0). 

PROP.  10.      TFAeft  a  plate  is  supported  at  two  edges,  and  a 
weight  applied  at  the  centre,  the  weight  of  the  plate  itself  not 
being  considered,  the  strength  is  constant  whatever  be  the  area. 
Let  the  plate  be  rectangular. 

FIG.  23. 


Take  the  line  p  p,  passing  -through  the  centre,  as  the  line 
of  fracture.  Call  the  sides  a  and  b.  Let  n  b  =  distance  to 
centre  of  gravity  of  each  half,  regarding  the  point  of  applica 
tion  of  the  weight  as  a  fulcrum. 


w  w  n 

Then,  -~-  .  n  b  —  a  R,  or  R  —  — 


— ,  which  is  a  con 


stant  as  long  as  the  ratio  of  b  to  a  is  constant. 

The  same  is  true,  if  the  fracture  be  supposed  to  take  place 
along  any  oblique  line,  for  if  the  plate  be  increased  or  dimin 
ished,  the  lines  w  o  and  A  B,  which  express  the  leverage  of  the 
weight  and  the  resistance,  will   always  bear  the  same  ratic 
(Fig.  25.) 

FIG.  24. 


When  the  weight  is  uniformly  distributed, 


n 


w  —  \  a  b,  w  x  n  b  =  J  n  a  b2  ~  a  R,  R  —  -$  t>2,  or  the 


48  BKIDGE   CONSTRUCTION. 

strain  is  proportional  to  the  area  of  the  plate.  Hence,  where 
there  is  no  applied  weight,  the  strength  of  the  plate  diminishes 
in  proportion  as  the  area  increases. 

KELATIVE  DEFLECTIONS. 

PROP.  11.     To  find  the  deflection  of  a  rectangular  beam 
supported  in  the  middle,  and  uniformly  loaded  over  its  length. 

FIG.  25. 


"    C  B 

Let  A  B  be  the  beam,  0  the  fulcrum,  x  =  distance  of  any 
perpendicular  (y]  from  the  extremity  B. 

When  the  wreight  is  at  the  extremity,  the  strain  upon  any 
section  will  be  as  the  distance  z,  and  will  be  represented  by 
w  x,  but  the  deflection  will  be  not  only  as  the  strain,  but  as 
the  distance  from  B;  hence,  it  will  be  proportional  to  w  x2,  or, 
ify  =  w  x2,  it  is  evident  that  y  corresponds  to  the  abscissa  of  a 
common  parabola  whose  ordinate  is  #,  and  the  whole  deflection 
equal  to  the  sum  of  these  abscissas  will  be  represented  by  the 
area  B  0  n'  =  J  rectangle  B  0  nf  R  =  (A). 

When  the  weight  is  uniformly  distributed,  the  strain  upon 
any  section  will  be  in  proportion  to  the  weight  and  distance 
from  B. 

w  x 
Let  x  be  any  distance,  then  I  :  w  :  :  x  :  —j-  =  weight  on 

the  part  x,  -~-y  .  x  =  moment  to  which  the  strain  or  exten- 

•+ 1 

sion  of  the  fibres  will  be  proportioned.     The  deflection,  being 

w 
as  the  strain  and  distance  from  B,  will  be  -^j  x3. 

If,  then,  y'  =  -~-j  #3,  we  perceive  that  yf  is  the  abscissa  of 
a  cubic  parabola  of  which  x  is  the  ordinate. 


RESISTANCE    OF    MATERIALS.  49 

The  area  BCn  =  l  of  rectangle  BCna  =  %BC n' E  = 
the  deflection. 

Hence,  the  deflection  in  the  two  cases  will  be  as  J  to  J,  or 
as  8  to  3.* 

PROP.  12.  The  deflection  of  a  beam  supported  at  the  ends 
and  uniformly  loaded  will  be  to  the  deflection  of  the  same  beam, 
when  the  whole  weight  is  in  the  centre,  as  5  to  8. 

FIG.  26. 


When  the  whole  weight  is  at  the  centre  let  w  represent  tho 
weight  upon  one  of  the  supports,  the  strain  upon  any  section  at 
the  distance  x  will  be  represented  by  w  x,  and  the  deflection, 
as  in  the  last  proposition,  by  w  x2.  It  will,  therefore,  as  in  the 
last  case,  correspond  to  the  abscissa  of  a  common  parabola,  of 
which  x  is  the  ordinate.  The  sum  of  these  deflections,  or  the 
whole  deflection,  will  be  proportional  to  the  area  Apne  =  j 
rectangle  Aon c. 

Let  the  beam  be  now  supposed  to  be  uniformly  loaded,  and 
let  the  deflection  due  to  the  extension  of  the  fibres  at  the  dis 
tance  x  be  ascertained.  It  is  evident  that  the  weights  upon 
the  points  of  support  will  be  the  same  as  formerly. 

The  reaction  of  the  point  A  may  be  represented  by  a  force 
equal  to  w  acting  upwards,  its  leverage  at  the  distance  x  will  be 
w  x,  and,  the  deflection  due  to  it,  w  x2,  as  before ;  but  the  effect 
of  the  uniformly  distributed  load  upon  the  part  x  diminishes 
this  deflection,  since  it  acts  in  the  opposite  direction  ;  its  effect 

*  Tredgold  gives  the  proportion  in  this  case  as  4  to  3  (see  treatise  on 
cast  iron,  page  141).  To  test  the  question  by  direct  experiment,  a  flexible- 
strip  of  wood  7  feet  long  was  suspended  at  the  middle.  Two  uniform- 
chains  of  the  same  length  were  laid  upon  the  top,  and  the  deflection 
found  to  be  ^  of  an  inch :  one  chain  was  then  suspended  at  each  end,, 
and  the  deflection  became  V  °f  an  *ncn  '>  hut,  4 :  11  : :  3  :  8£,  a  result 
much  nearer  the  calculated  proportion  than  wa*  sxpected  with  the-  appa 
ratus  used. 
4 


50  BRIDGE   CONSTRUCTION. 

W 


will  be  <r;  £3?  and  the  whole  deflection  will  therefore  be  \w  x 

—  27  xr     "^ie  exPressi°n  97 ^3  *s  represented  by  the  area  A 

p'  n'  <?,  which  we  have  already  shown  to  be  J-  rectangle.     And 
hence,  the  deflections  will  be  as  j-  —  J- :  J,  or  as  5  to  8. 

STRENGTH  OF  PARTICULAR  SECTIONS. 

PROP.  13.     Strength  of  a  triangular  section. 

As  this  case  is  more  curious  than  useful,  we  will  simply 
indicate  the  mode  of  procedure  without  entering  into  its  full 
investigation. 

FIG.  27. 


Let  ABO  represent  the  section  at  the  point  of  greatest  strain 
h  =  height  and  b  —  base  of  triangle,  ppf  neutral  axis 
R  —  the  maximum  strain  upon  a  superficial  unit. 
The  strain  varying  as  the  distance  from' the  neutral  axis,  it 
will  be  =  R  at  the  point  A,  and  at  any  point  of  B  0  it  will  be 

Rx 

found  thus,  (h  —  x) :  x  : :  R  :  T •     The  strain  upon  the 

fi  -     x 

upper  part  of  the  section  will  be  represented  by  a  pyramid, 
whose  base  is  A ppf,  and  altitude  R ;  upon  the  lower  part,  it 
will  be  the  wedge-formed  solid,  whose  base  is  p  p1 B  C,  and 

altitude  -^ —  — .     The  volume  of  the  solid  will  be  the  difference 
h  —  x 

between  the  wedge  pp'r  B  0  and  pyramid  pf  p"  C. 

By  equating  the  moments  we  obtain  the  value  of  xy  and, 
consequently,  the  position  of  the  neutral  axis :  this  value  sub 
stituted  in  the  expression  for  the  resistance  of  the  section  will 
give  its  value  in  terms  of  b  and  h. 


RESISTANCE    OF   MATERIALS.  51 

It  is  found,  that  the  strength  of  the  triangular  prism,  is  to 
that  of  a  rectangular  prism  having  the  same  base  and  altitude 
as  339  :  1000,  or  nearly  as  1  :  3. 

As  the  resistance  to  compression  and  extension  are  supposed 
equal,  the  prism  must  be  equally  strong,  whether  the  base  or 
vertex  be  compressed,  provided  the  limit  of  elasticity  be  not 
exceeded. 

The  investigation  of  this  case  leads  to  an  apparently  para 
doxical  result  :  it  is  found,  that  the  prism  becomes  one  thirty- 
seventh  part  stronger  when  the  angle  is  taken  off  to  one-tenth 
of  the  depth. 

The  difficulty  will  vanish,  when  it  is  remembered  that  the 
greatest  resistance  of  any  fibre  is  jR,  and  that  in  the  triangular 
section,  only  the  single  point  at  the  apex  opposes  this  resist 
ance  ;  whereas,  if  a  portion  be  removed,  every  point  of  the  line 
n  nr  opposes  a  resistance  =  R. 

The  triangular  section  contains  half  the  surface  of  the  cir 
cumscribing  rectangle,  but  is  only  one-third  as  strong  ;  hence, 
there  is  no  economy  in  its  use. 


PROP.  14.     The   strength   of  a  cylinder  supported  at  the 

3*- 
ends  is  to  that  of  its  circumscribed  prism  as  -TT  :  1,  or  as  -589 


FIG.  28. 


The  neutral  axis  being  at  the  centre,  the  resistances  to 
compression  and  extension  will  be  represented  by  the  ungulas. 
whose  bases  are  the  semicircles  An  B  and  A  nf  B. 

n 

The  volume  of  the  ungula  has  been  found  to  be  2  r2  -Q-  . 


52  BRIDGE   CONSTRUCTION. 

The  distance  of  the  perpendicular  through  the  centre  of 
gravity  is  T3g  it  r.* 

*  To  find  the  volume  and  the  position  of  the  centre  of  gravity  of  an  un« 
gula  or  solid  formed  by  passing  an  oblique  plane  through  the  diameter 
of  the  base  of  a  semi-cylinder. 

FIGS.  30  and  31. 


All  the  sections  parallel  to  A  B  -will  be  rectangles,  the  altitudes  of 
which  will  be  proportional  to  their  distance  from  C. 
Hence,  if  R  represent  the  altitude  at^>,  and  o:  =  the  distance  of  any 

Rx 

section,  r  =  radius,  y  =  ordinate,  r  :  x  : :  R  :  — —  =  perpendicular  of 

R 

rectangle,  and  2  —  x  y  =  area. 

The  elementary  solid  will  be 2  —  xydx,  and  its  moment  =  2  —  x-  y  d x» 
The  distance  to  the  centre  of  gravity  will  be  -^ — \~^~x' 


1.  To  determine  the  volume  of  theungula  we  have  y=\/  r2  —  x2;  hence, 
fxydx=f(r*  —  x*}^xdx.     Maker2  —  x3  =  z*.     Whence,  xdx=—  z 


__ 

which  becomes,  when  taken  between  the  limits,  o  and  r,  =  —  5- 

o 

This  negative  result  does  not  effect  the  absolute  volume  :  to  interpret 
it,  it  must  be  observed  that  the  integral  does  not  become  o  when  x  =  o, 
but  when  x  —  r,  anl  consequently  the  solid  lying  in  the  direction  of  C 

2R    r3      27?r2 
from  p  should  be  negative.     —  —  *  -«r  =  —  ^  —  =  volume  required  by 

/»2J2 

substitution  in  the  expression   i  -  xydx. 


RESISTANCE   OF   MATERIALS.  53 

Hence,  the  moment   becomes   2  rl  •  -~  •  -f$  *  r  =  — ~ — • 

R 

The  moment  of  the  rectangle  Ap  is  2  r2 '  -7  •  f  r  —  f  R  r3. 

3tf 
But  J.ftr3:  J-ff^r3::  1  :  -r  :  :  1  :  589. 


The  volume  of  the  ungula  is  therefore  equal  to  that  of  a  pyramid  whos$ 
base  is  the  circumscribing  rectangle  with  the  same  altitude. 

To  find  the  centre  of  gravity,  we  have^a;2  y  dx  =f(r2  —  x2}*  x2  dx=- 
i  -  (r2  —  x2)*  2  x  dx.     Integrate  by  parts,  making  in  the  formula 

fzdy  =  zy—fydz        z  =  ^        y  =  (r2  —  x*fi 
whence,  d  y  =  f  (r2  —  z2)  2  z  d  z.     Substitute  these  values  we  obtain  I  7; 


—x2)dx.     The  quantity 

3  x 
(r2  —  x2)  2  ^  will  reduce  to  o  when  x  =  o  or  x  =  r,  this  term  wilL  therefore 

disappear  and  the  expression  reduces  tofxz(r2  —  x*)^dx  =  i/X?'2  —  x2)^ 


x2  dx.    Transpose  the  last  term  to  the  first  number,  and  reduce/(r2  —  a;a)  * 


But  the  integral/(7<2  —  x2}'2  (d  x~),  between  the  limits  o  and  r,  represents 
the  area  of  a  quadrant  =  4  rt  r2. 


=—^-==T3F7t?*  =  distance  from  the  centre  of  the  circle  to  the  perpendicular 

3 

through  the  centre  of  gravity. 

The  line  which  joins  C  and  the  middle  point  of  R  passes  through  the 
centres  of  all  the  elementary  rectangles,  and,  therefore,  the  centre  of 
gravity  must  be  found  at  the  intersection  of  this  line  with  the  perpen 
dicular,  through  a  point  at  a  distance  of  T\  rt  r  from  the  centre.  Its 
dic-tance  from  the  base  is  therefore  the  fourth  term  of  the  proportior 


54 


BRIDGE   CONSTRUCTION. 


PBOP.  15.  If  n  represent  the  ratio  of  the  inner  and  outer 
diameters,  the  strength  of  the  solid  cylinder  will  be  to  that  of  a 
tube  of  the  same  exterior  diameter  as  1 :  (1  —  n*). 


Let  r  —  exterior  radius,  n  r  =  interior  radius,  R  =  strain  at 
Ay  then  r  :  nr  :  :  R  :  n  R  =  strain  at  P. 

*  r*  ]l 
The  resistance  of  the  semicircle  0  A  is  —  «—  .     (See  last 

problem). 

tfn4r37t 
The  resistance  of  the  semicircle  OP  is  --  x  --  . 

3    7L) 

The  resistance  of  the  ring  is  —  —  (1  —  n4) 


PROP.  16.     To  find  the  strength  of  a  vertical  rib  with  he* 
rizontal  flanges  on  both  sides. 

FIG.  32. 


This  case  can  be  immediately  deduced  from  that  of  a  rect 
angular  section,  for  the  area  is  evidently  equal  to  the  rectangle 
A  0 —  2  rectangles  n  m,  and  as  the  neutral  axis  is  in  the  cen 
tre,  the  strength  will  be  equal  to  the  difference  of  the  strength 


RESISTANCE   OF   MATERIALS.  55 

of  these  rectangles.     Call  b  d  =  the  circumscribing  rectangle, 
A  C,  bf  df  the  deducted  rectangles  2  n  m. 

But  the  strain  on  the  extreme  fibres  of  the  inner  rectangle 
is  not  as  great  as  R  ;  it  is  therefore  necessary  to  introduce  the 
relative  values  of  these  strains.  The  strain  at  the  distance 
A  df  from  the  neutral  axis  is  determined  from  the  proposition 

" 


d 
b  d2  R  represents  the  resistance  of  the  rectangle  ¥  d' 


Cv 


=  resistance  of  the  rectangle  b'  df 


b'  d'3 

These  resistances  are  to  each  other  as  b  d2  :  -  -  —  :  :  b  d'* 

d 

:b'd'3. 

Hence,  the  strength  of  the  circumscribed  rectangle  is  to 
that  of  the  given  section  as  b  d3  :  (b  d3  —  V  dry). 

PROP.  IT.  The  strongest  beam  that  can  be  cut  out  of  a 
tree  or  given  cylinder  has  the  breadth  and  depth  in  the  pro 
portion  of  1  to  V2. 

FIG.  83. 


The  strength  is  as  b  d2,  or  as  the  product  of  the  breadth 
and  square  of  the  depth.  Call  d'  =  diameter  of  cylinder,  x  = 
depth ;  then,  Vd'2  —  x 2  =  breadth,  and  x2  Jd'2  —  x2  =  a  max 
imum.  The  first  differential  co-efficient  is  4td'*x3  —  6  x5, 
placing  this  equal  to  zero,  x  becomes  =  d'  \/| ;  substitute  this 
value  of  x  in  the  expression  for  the  breadth,  it  becomes 
jd'2  —  %d'*  =  dr  VI 

But  d'  V§  :  d'  Vi  :  :   V2  :  1. 

In   precisely   the   same  way,  it    can   be   shown,  that   the 
stiffest   beam   which   is   proportional    to   b  d3   will   have   its 


56 


BRIDGE   CONSTRUCTION. 


breadth  to  its  depth  as  1  :   v/3.     In  this  case  the  breadtt 
is  equal  to  the  radius. 

The  geometrical  construction  of  the  figure  of  the  stiffest 
beam  is  extremely  simple.  From  opposite  extremities  of  any 
diameter  with  radii  equal  to  the  radius  of  the  cylinder  describe 
arcs  cutting  the  circumference  and  join  the  points  of  intersection. 

FIG.  34. 


The  construction  of  the  figure  of  the  strongest  beam  is  also 
very  simple,  for,  since  the  sides  are  as  1  :  \/2,  the  hypothenuse 
will  be  V  3,  and  from  the  properties  of  right-angled  triangles, 

v/3  :  1  :  :  1  :  AC=~;  but,  — L :  v/3::l  :  3;  hence, 
s/o  >/  o 

AC=IAB. 

Lay  off  therefore  one  third  of  the  diameter,  and  ere'ct  a 
perpendicular ;  its  intersection  with  the  circumference  will  de 
termine  the  point  D. 


PROP.  18.     To  find  the  resistance  of  a  beam  lying  hori< 
zontally  upon  an  edge. 

FIG.  36. 


Let  A  B  be  a  diagonal  h  =  perpendicular  OP  R  = 
strain  upon  (7,  the  whole  strain  will  be  represented  by  the 
pyramid,  whose  base  is  A  B  0  and  altitude  R ;  its  volume  is 


RESISTANCE   OF   MATERIALS.  57 

I  d    R       I  dR       „,      -i-  ,  i 

~~v~~  '  T  =     —  fi"  —  "  distance  from  A  B  to  the  perpendic' 


ular  through  the  centre  of  gravity  is  ^  A,  and  the  moment  wi 
therefore  be  —  —  —  .     The  moments   of  the   two  pyramids 

4  .S  (7  and  4  -BD  will  be   ldhR 


6 

When  the  side  A  0  is  vertical  the  moment  is : 

6 

The  ratio  of  the  strength  will  therefore  be  as  d  :  h,  or  as 
the  side  is  to  the  perpendicular. 

When  the  section  becomes  a  square  we  have  d  :  h  :  :  V%  : 
1  ;  hence,  the  strength  of  a  square  beam  where  the  side  is 
vertical  is  to  the  strength  when  the  diagonal  is  vertical  as 
s/2  :  1. 

PROP.  19.  When  the  pressure  upon  a  beam  supported  at 
the  ends  varies  as  the  distance,  the  point  of  greatest  strain  will 
be  at  a  distance  from  the  unloaded  extremity  equal  to  tho 
length  multiplied  by  the  square  root  of  one-third,  or  (I 

FIG.  36. 


In  this  case,  the  pressure  will  be  proportional  to  the  area 
of  a  triangle,  the  centre  of  gravity  of  which  is  at  a  distance 
from  A  =  1 1. 

Then,  for  the  weight  upon  A  we  have  /:-—::  w  :  --  = 

o  o 

weight  at  A. 

In  like  manner  —^-  =  weight  at  B. 
o 

Let  x  =  distance  of  any  section  from  A.  The  resistance 
of  the  support  A,  being  regarded  as  a  force  =  -^-  acting  up- 


58  BRIDGE   CONSTRUCTION. 

wards,  will  be  opposed  by  the  weight  upon  x  acting  downward, 
and  the  difference  of  the  moments  will  represent  the  strain. 

For  the  first  we  have  —  -  •  x  =  -—-  =    moment   of   force 
o  o 


For  the  second  we  have,  since  the  weights  will  be  as  the 

7/J  ^7* 

square  of  the  lengths,  I2  :  x2  :  :  w  :  ——-  =  weight  on  x.     As 

I 

0.1  .      X       W  X2  WX  c      ,, 

the    leverage   is  -     —  ^—  •  I  x  =  ——  -    =    moment    of    the 

O  I  O  I 

weight  on  x. 

The  difference  will  be  (^-  —  ^TT\  =  strain  on  section. 
\    o  o  I    / 

By  the  principles  of  maxima  and  minina  we  have 


PROP.  20.  To  determine  the  extension  of  the  fibres  when 
a  learn  is  supported  at  the  ends  and  loaded  in  the  middle. 

A  beam  supported  at  the  ends  and  loaded  in  the  middle  is 
in  the  same  condition  as  a  beam  resting  upon  a  fulcrum  in  tho 
middle  and  loaded  with  equal  weights  at  the  ends. 

FIG.  37. 

i ...  ,-|j 


jy 


Let  I  =  one-half  the  whole  length 
w  =  the  weight  on  A 

e  —  the  maximum  extension,  which  will  be  at  C. 
Now,  as  the  extension  at  any  distance  is  in  proportion  to 
the  strain,  it  will  evidently  be  in  proportion  to  x  ;  and  we  have 

e  x 
therefore,  I  :  x  : :  e  :  — - —  =  extension  at  the  distance  x. 

I/ 

*  Tredgold  gives  it  vAp. 


RESISTANCE   OF    MATERIALS.  59 

But,  the  deflection  being  as  the  extension  and  distance  di- 

festly,  and  inversely  as  the  depth,  it  will  be  as  — — .  — =  - — . 

'       d       Id 

P 

Call  this  expression  (y),  we  have  therefore,  y  —  - — #2=the 

/  d 

equation  of  a  parabola,  of  which,  x  is  the  ordinate,  and  y  the 
abscissa. 

The  whole  deflection  being  equal  to  the  sum  of  these  ab 
scissas  will  be  represented  by  the  area  A  02)  =  ^  rectangle 

A  D  =  -II  -  (~ 12)  =  e£.     The  deflection  of  the  part  B  0 
V  d     '       o  d 

2  e  I2 

being  equal  to  that  of  A  (7,  the  whole  deflection  will  be  — — . 

3  d 


,T71             3  d  X  (deflection) 
Whence,  -^-^ '-  =  e. 

—  / 


By  observing  the  deflection  produced  by  a  given  weight, 
and  substituting  its  value  in  the  above  expression,  the  value 
of  e  can  be  ascertained.  For  cast  iron,  when  the  weight  is 
15,300  Ibs.  per  square  inch,  it  is  found  to  be  7^4  inches  for 
a  length  of  one  inch. 

Means  of  determining  the  constants. 

B  .D3 

The  equation  H'  —  -  —  -  expresses    the   relation   between 
iv  lz 

the  dimensions  of  a  beam  when  the  deflection  is  in  proportion 
to  the  length.  If  this  deflection  be  assumed  at  ^  of  an  inch 
for  every  foot  of  length,  and  d  =  the  observed  deflection 
caused  by  the  application  of  a  weight  w. 

Then,  d  :  w  :  :  —  :  —  —  —  =  weight  required  to  produce  the 
given  deflection.  By  substituting  this  value  for  w,  we  obtain 


To  determine  the  value  of  R,  experimentally,  let  a  beam 

*  This  expression  is  the  same  as  that  given  by  Tredgold,  but  the 
manner  of  obtaining  it  is  far  more  direct  and  simple.  (See  Treatise  on 
sast  iron,  p.  136.) 


60  BRIDGE   CONSTRUCTION. 

be  placed   upon   two   supports   and   loaded  with  any  known 
weight,  not  so  great  as  to  impair  the  elasticity. 

Observe  the  deflection  (d),  the  weight  (w\  the  distance  be 
tween  the  supports  (Z),  the  breadth  (#),  and  the  depth  (D).  The 
length  being  in  feet,  and  the  other  dimensions  in  inches. 

Substitute  these  values  and  perform  the  operations  indica 
ted,  the  value  of  R  vill  be  obtained. 

This  constant  for  cast  iron  has  been  found  to  be  -001    Tredgold. 
"         «         White  fir       "         "         "     -01  " 

"         "         Oak  "         "         "     -0109        " 

«         "         Yellow  pine  "         "     -0115        " 

"         "         American  white  pine        "     '0125  Author. 
The   formula  which  expresses   the   strength  of  a  beam  is 

Q         7  * 

• =  II  when  the  beam  is  supported  at  both  ends  and  the 

2bd2 

weight  applied  in  the  middle. 

To  determine  the  constant,  weights  should  be  applied  and 
gradually  increased  S3  long  as  no  perceptible  flexure  remains 
upon  their  removal. 

The  highest  value  of  w  thus  obtained  will  give  the  value 
of  R.  In  this  formula,  R  expresses  the  maximum  strain  upon 
a  square  inch ;  but,  in  determining  its  value,  when  used  in 
proportioning  the  parts  of  important  structures,  it  is  proper 
to  observe,  that  the  strength  of  materials  generally  dimin 
ishes  as  the  length  of  time  in  use  increases,  and,  that  a 
weight  which  will  produce  no  perceptible  deflection  in  a  short 
time,  may  produce  a  very  great  deflection  when  long  continued. 

From  some  experiments,  made  by  the  writer  in  the  spring 
of  1840,  it  appeared,  that  locust  would  bear  for  a  few  seconds 
a  strain  of  5500  pounds  per  square  inch  without  apparent 
injury,  but  the  elasticity  was  impaired  by  2304  pounds  per 
square  inch  continued  16  days. 

The  value  of  R  for  cast  iron  when  the  time  was  short  was 
found  by  Tredgold  to  be  15300  pounds 

For  White  fir  the  constant  is  3519       " 

"    Oak  "         "         "  3825       " 

"    Yellow  pine         "         "  3825       " 

The  above  constants  were  deduced  from  experiments  and 


RESISTANCE   OF    MATERIALS. 


61 


other  data  furnished  by  Trcdgold  in  his  treatise  on  cast  iron, 
but  the  writer  believes  them  to  be  entirely  too  great  for  perma 
nent  strains  :  in  the  course  of  experiments,  already  referred  to,* 
he  found,  that  when  white  pine,  yellow  pine,  and  hemlock, 
were  subjected  to  a  strain  of  from  1500  Ibs.  to  1800  Ibs.  per 
square  inch,  continued  for  16  days,  the  pieces  did  not  recover 

*  These  experiments  were  made  at  York  in  the  year  1840.  The  writer 
upon  commencing  his  duties  as  an  engineer  on  the  York  and  "Wrightsville 
Rail  Road,  found  in  the  office  a  number  of  very  fine  specimens  of  wood,  that 
had  been  procured,  for  the  purpose  of  experimenting  upon  them,  by  T. 
Jefferson  Cram,  formerly,  assistant  professor  of  natural  philosophy  at 
the  U.  S.  Mil.  Acad.,  and  at  that  time,  a  civil  engineer  on  the  Baltimore 
and  Susquehanna  Rail  Road.  Unwilling  to  lose  so  favorable  an  oppor 
tunity,  several  other  specimens  were  added  to  the  number,  and  experi 
ments  made  with  a  view  of  determining,  not  the  absolute  strength,  but 
the  elastic  limit.  The  result  is  given  in  the  following  table : 

When  the  pieces  did  not  exactly  recover  their  shape  they  are  marked 
injured. 


Kind  of  Timber. 

Strain  per  sq.  inch. 

Time. 

Remarks. 

1 

White  pine 

2272 

10  min. 

Injured. 

2 

do. 

1548 

16  days 

Injured. 

3 

Hemlock 

2624 

5  min. 

Injured. 

4 

do. 

1620 

16  days 

Injured. 

5 

Yellow  pine 

2848 

5  min. 

Injured. 

6 

do. 

1800 

16  days 

Injured. 

7 

Locust 

5504 

2  min. 

Not  injured. 

8 

do. 

3600 

3£  days 

Injured. 

9 

do. 

2304 

16   do. 

Injured. 

10 

White  oak 

4248 

15  min. 

Not  injured. 

11 

do. 

7200 

do. 

Injured. 

12 

do. 

3648 

40  hours 

Not  injured. 

13 

do. 

4088 

48    do. 

Injured. 

The  pieces  were  all  5  feet  long,  3  inches  deep,  1  inch  wide,  supported 
at  one  foot  from  the  end.  The  weight  acting  with  a  leverage  of  4  feet. 

R  is  determined  from  the  formula  R  =  ^—j^. 

o  a3 

There  were  three  pieces  of  each  kind,  all  very  superior. 

From  these  experiments  it  appears,  that  there  is  a  great  difference  in 
the  powers  of  resistance  of  different  kinds  of  timber,  and  that  Oak  and 
Locust  are  far  superior  to  Hemlock  and  Pine.  Also,  that  a  small  weight 
long  continued  may  produce  more  permanent  flexure  than  a  much  greater 
one  applied  only  for  an  instant. 


62  BRIDGE   CONSTRUCTION. 

their  shape  upon  the  removal  of  the  weight ;  and  in  practice, 
he  has  not  considered  it  safe  to  assign  more  than  800  Ibs.  per 
square  inch  as  a  permanent  load  and  1000  as  an  accidental 
load. 

The  following  table  may  be  found  of  much  utility : 
Column  A  contains  the  constants  used  in  the  formula  for 
the  stiffness  of  beams  supported  at  the  ends  and  loaded  in  the 

T>  JH3 

middle.     R  =  — r^,  (dimensions   all   in  inches,  except  the 

length,  which  is  in  feet.) 

Column  B  contains  the  constants  used  in  the  formula  for 

the  strength  R  —  oT~T2  =  strain  on  a  square  inch  in  pounds, 

(the  dimensions  being  all  in  inches.) 

Column  C  gives  the  greatest  extension  without  injury. 

"       D         "        specific  gravity. 

"       E         "        weight  of  a  cubic  foot  in  Ibs. 

"       F  wt.  of  the  modulus  of  elasticity  in  Ibs. 

"       (JT          "        height  of  the  mod.  of  elasticity  in  feet. 
The  data  have  been  obtained  from  Tredgold. 


A 

B* 

c 

D 

E 

F 

G 

Cast  iron 

•001 

15,300 

T2W 

7-2 

450 

18,400,000 

5,750,000 

Malleable  iron 

•0008 

17,800 

iVoff 

7'2 

475 

24,920,000 

7,550,000 

White  fir 

•01  f 

3,630 

Tff  i 

0-47 

29-3 

1,830,000 

8,970,000 

Oak 

•0109 

3,960 

l£l 

0-83 

52 

1,700,000 

4,730,000 

Yellow  pine 

•0105 

3,900 

*h 

0-46 

26| 

1,600,000 

8,700,000 

The  weight  of  the  modulus  of  elasticity  is  determined  by 
the  proportion  1  :  -  :  :  w  :  modulus,  (e)  is  found  in  column 
C,  and  (w)  in  column  B. 

*  The  constants  in  column  B  I  consider  too  great  for  timber ;  as  re 
gards  the  others  I  can  express  no  opinion,  having  made  no  experiments. 
As  a  general  rule,  I  should  not  think  it  safe  in  practice  to  use  higher 
numbers  than  those  given  in  column  B  divided  by  four. — Author. 

f  By  experiments  of  Author  on  American  white  pine  the  constant  is  '0125. 

O^T"  In  applying  the  above  formulas  it  must  be  observed,  that  w  includes 
the  weight  of  the  beam  itself.  To  find  the  weight  sustained  at  the  middle 
point,  one-half  of  the  weight  of  the  beam  must  be  deducted ;  or,  if  the 
load  is  uniformly  distributed,  deduct  the  whole  weight  of  the  beam  and 
multiply  by  2. 


WOODEN  BRIDGES. 


INSTEAD  of  commencing  this  treatise  with  a  history  of  Bridge 
Construction,  and  an  explanation  of  the  various  plans  that 
have  from  time  to  time  been  adopted,  it  is  believed  that  a 
preferable  mode  will  be  to  establish  first  the  true  principles  of 
construction,  and  then  to  apply  these  principles  to  an  exami 
nation  of  plans  that  have  been  executed.  By  this  means  it 
can  be  ascertained  how  far  they  coincide  with,  or  depart  from, 
the  principles  which  we  endeavor  to  establish,  and  how  far 
the  correctness  of  these  principles  is  confirmed  by  practical 
experience. 

The  most  important  part  of  any  bridge,  and  that  which 
admits  of  the  greatest  variation  in  form  and  principle,  is  the 
support  of  the  roadway.  To  this  therefore  our  chief  attention 
will  be  directed. 

The  most  simple  support  for  a  roadway  evidently  consists 
of  a  series  of  longitudinal  timbers  laid  between  two  abutments 
or  piers.  And  as  the  examination  of  this  case  will  lead,  by 
easy  gradations,  to  others  which  are  more  complicated,  and  as 
it  also  involves  many  of  the  principles  which  apply  to  struc 
tures  of  a  more  important  character,  we  will  begin  by  an 
examination  of  the  forces  which  act  upon  a  single  beam  laid 
upon  two  supports  and  loaded  with  a  weight,  either  uniformly 
distributed,  or  concentrated  at  any  given  point. 


64  BRIDGE   CONSTRUCTION. 

It  has  been  shown  in  treating  of  the  resistance  of  solids, 
that  the  fibres  on  the  upper  side  will  be  compressed,  and  on 
the  lower  side  extended ;  that  within  the  elastic  limits  the  re 
sistances  to  these  forces  are  equal ;  that  the  intensity  of  the 
strain  varies  directly  as  the  distance  of  any  fibre  from  the  neu 
tral  axis,  and  that  at  the  axis  itself  the  strain  is  nothing. 

There  exists,  also,  a  force  called,  by  Dr.  Young,  detrusicn, 
the  effect  of  which  is  to  crush  across  the  fibres  close  to  a  fix^d 
point,  and  the  resistance  to  which  is  directly  proportional  to 
the  area  of  the  cross  section. 

This  force,  a8  has  been  shown,  modifies  the  form  of  a  beam 
of  equal  strength,  which,  instead  of  being  the  apex  of  a  conic 
section  at  its  extremity,  must  be  enlarged  sufficiently  to  resist 
this  force  of  detrusion. 

The  existence  of  this  vertical  force,  and  its  effects  at  other 
points,  have  not  been  considered  by  writers  on  the  resistance 
of  solids,  probably  because  it  diminishes  rapidly  in  approach 
ing  the  centre  of  abeam,  whilst  the  area  of  the  section  generally 
increases.  That  a  vertical  strain  upon  the  fibres  exists  at  other 
points  can  be  shown  by  the  following  considerations. 

Let  A  B  represent  a  beam  supported  at  A  and  B,  and  dis 
regarding  for  the  present  its  own  weight,  let  it  be  loaded  with 
a  weight  applied  at  the  centre. 

FIG.  38. 

s  „ 


This  force  is  directly  transmitted  to  the  points  A  and  B, 
each  of  which  sustains  one  half  the  weight.  The  lines  of 
direction  of  the  forces  are  along  A  W  and  B  W. 

By  constructing  the  parallelograms  of  forces  on  the  diago 
nals,  we  find  w  o  =  \w  n  =  ^w  for  the  vertical  forces  trans 
mitted  to  A  and  B,  and  p  o  =  the  horizontal  strain  at  w9  which 


J~Li  i.  L  t  V  Vt       tV/    -*--•-     WAJLVt.    -*--'  5     CU.li.V-*.      *S     IS     UXJLV^      fllitW*  AJUW.U  VVM.      0VAUWAA      Ct  b       i 

is  determined  by  the  proportion  d  :JZ : :  ^w  :  o  p  = 


'   ',  (in 


WOODEN   BRIDGES.  65 

which  /  represents  the  length,  and  d  the  depth.)  The  same 
force  is  transmitted  to  B.  We  can  also  determine  this  horizon 
tal  force,  by  the  condition  that  it  shall  keep  the  part  w  c  in 
equilibrio.  Regarding  w  as  a  fulcrum,  and  the  weight  at  B  = 

w  I 
J  w,  the  moment  of  this  force  will  be  J  w  x  J  I  =  —r~.     The 

moment  of  the  horizontal  force,  acting  with  a  leverage  d,  will 

w  I 

be  Ifdj  and  H '=  -r-j  as  before. 
4  d 

We  will  now  consider  the  action  of  these  forces  at  another 
point  ($),  the  weight,  as  before,  being  applied  entirely  at  the 
middle  point  of  the  beam. 

1.     Horizontal  strain  at  S. 

Since  the  weight  w  is  equally  supported  by  each  of  the 
points  A  and  J5,  we  may  continue  to  consider  (w)  as  a  fulcrum, 
and,  that  forces  (J  to),  acting  upwards  at  A  and  JB,  maintain 
the  equilibrium. 

The  portions  (A  n)  and  (n  B)  will  be  in  the  condition  of 
beams  fixed  at  one  end,  and  loaded  at  the  other. 

The  weight  J-  w  applied  at  B,  acting  with  a  leverage  u  =  /S 

0  produces  an  effect  equal  to  the  product  -^  X  u,  and  the  hori 
zontal  strain  at  S  acting  with  a  leverage  d  has  for  its  moment 

w  u  u  w 

H  x  d.      Hence,  H  d  =  —^-  or  H  »=  -5-%,  which  becomes 

/  I  w 

when  (u  —  T>),  w  =  j-,,  as  before. 

The  horizontal  strain  in  the  middle  of  the  beam  is  to  the 
same  strain  at  any  other  point  as  -^  :  u,  and  consequently  va 
ries  with  the  perpendiculars  of  a  triangle  constructed  on  -$  aa 
a  base. 

2.     Vertical  force  at  any  point. 

f\i   rt/t 

The  horizontal  at  S  was  found  to  be  --j,  but  it  is  evident, 


66  BRIDGE   CONSTRUCTION. 

that  the  portion  of  the  beam  D  $,  with  the  applied  weight  at 
W7  presses  against  the  cross  section  at  $,  and  must  be  resisted 
by  the  reaction  at  that  point. 

u  w 
If  the  horizontal  force  -^—7,  acting  with  its  leverage  (d), 

was  sufficient  to  sustain  the  part  D  S,  the  effect  of  the  weight 
at  W  would  be  entirely  overcome,  and  there  would  remain 
nothing  to  produce  a  downward  strain  upon  the  fibres  at  ($), 
or,  in  other  words,  the  vertical  force  would  be  zero.  That 
this  is  not  the  case,  however,  can  be  seen  by  estimating  the 
force  necessary  to  sustain  D  >S  in  equilibrium. 

As  ( W)  acts  with  a  leverage  D  W  or  A  n,  the  equation  of 

nij     1  ni\     7 

moments  will  be  Hf  d  —  -77-,  or  Hf  =  5—,.     But  we  have 

-  Ad 

eeen  that  the  horizontal  strain  at  S  is  actually  H  =  Q—J. 

w 
The  difference  is  (Hr  —  H)  =  ^(l  —  u). 

As  this  expression  cannot  become  zero  for  any  point  be 
tween  W  and  C,  it  follows,  that  the  horizontal  force  is  not 
sufficient  to  sustain  the  weight,  and  there  must  consequently 
be  a  cross  strain  upon  the  fibres  which  must  compensate  for 
this  deficiency,  and  be  resisted  by  a  vertical  reaction. 

Call  this  vertical  force  /,  it  acts  with  a  leverage  =  D  S  = 
(I  —  u).  The  difference  of  the  horizontal  forces,  or  II'  —  H, 
acts  with  a  leverage  =  d.  The  equation  of  moments  will 

w 
therefore  be  /  (I  —  u)  =  d  .  5-3  (I  —  u\  from  which,  we  ob- 

w 
tain/—  -fT,  or,  the  cross  strain  upon  the  fibres  produced  ~by  a 

weight  applied  in  the  middle  is  constant  at  every  point,  and 
equal  to  one-half  the  weight. 

We  can  obtain  the  same  result  by  another  method.  Using 
the  same  notation  as  before,  we  may  suppose  that  the  vertical 
force  (/)  [acting  at  S  with  a  leverage  SJ)  =  (I  —  u),']  and  the 
horizontal  strain  at  jS,  acting  with  a  leverage  d,  sustain  in 

equilibrium  the  weight  (W)  acting  with  a  leverage  W  D  =  -9-. 


WOODEN   BRIDGES.  67 

The  equation  of  moments  will  be 


Substitute  the  value  of  H=  —  -  and  reduce  we  find/  =  -—  as 

L  (/  *j 

before. 

Again,  let  (  W)  be  a  fulcrum  and  (^  w)  a  force  acting  up 

wards  at  By  with  a  leverage  W  0  =  —  ,  W  being  taken  as  the 

2 

point  of  rotation.    This  force  will  be  resisted  by  the  horizontal 
force  acting  with  a  leverage  (d),  and  /  acting  with  a  leverage 

(|—  «)=TT-sr. 

The  equation  of  moments  mil  be 


which,  by  reduction,  gives  /  =  —  ,  as  in  the  other  cases. 

When  the  weight  is  not  applied  in  the  middle,  it  may  be 
shown  in  the  same  way,  that  the  vertical  forces  on  each  side 
will  be  constant  and  equal  to  the  pressures  on  the  points  of 
support. 

Let  the  weight  be  uniformly  distributed. 

In  this  case  the  forces  will  be  determined  in  two  different 
ways,  as  this  will  serve  to  verify  the  results,  and  exhibit  more 
fully  the  manner  of  their  action  and  distribution. 

1.  By  means  of  the  moments. 

The  weight  being  uniformly  distributed,  one-half  of  it  will 
be  sustained  by  each  point  of  support.  To  estimate  the  strain 
at  any  point,  $,  we  may  suppose  the  part  D  S  to  be  fixed,  and 
the  weight  \  w  at  .5,  to  act  upwards  with  a  leverage  u,  this 
force  is  opposed  by  the  weight  of  the  portion  u  acting  down 

wards  with  a  leverage  ^  ;  hence,  if  W  =  strain  at  S,  II'  d  — 
2 

W  U          U2W          .          UW       1  -  «  ,        .  ,  .  ™ 

=  horizontal   strain  at  & 


TT.  - 

^-y-,  II'  =  —  —  ,  /  —  -  —  \  = 
A  I  2  d     \     I     / 


when  u  =  —  we  have  If  =  —  —  —  =  horizontal  strain  at  centre. 
2  o  d 


68  BRIDGE   CONSTRUCTION. 

2.  By  resolution  of  forces. 

The  weight  at  $  is  one-half  of  the  portion  on  j$  2)  and 
one-half  of  that  on  S  (7,  consequently,  it  is  equal  to  one-half 

IV 

of  the  whole  weight,  or  — .     Join  S  A  and  SB,  and  let  Sn* 

<L 

=  2  w.     From  similar  triangles  we  can  obtain 
AB  :  An1  ::  Sn'  :  So' 

j        ,  W          a    ,         W    I U 

or     l:l-u::-:So>- 


2  21 

=  weight  transmitted  to  B  from  S9 

and  - —  (/  —  u)  +  (-—j  u  =  ^  the  weight  on  u)  =  -— ,  as  it 

1.   /  £  v  lj 

should  be,  for  the  whole  weight  at  B. 

We  also  have  S n'  :  n' B  :  :  So'  :  o' m  =  * ?Q' '*? '  B  = 

A  I)        [7      .„ r      7/|  7/      01)  /     j .      fit 

^       — -  X  u  —  —  ( — - — \  for  the  horizontal  force  at  S,  as 

before. 

The  same  results  would  be  attained,  for  mf  o"  =  the  hori 
zontal  force  in  the  direction  Sd. 

The  horizontal  force  that  would  sustain  the  part  tS  C,  as 

„          11,1-         ,1  •      U  W          U          1  U  W      U 

tound  by  taking  the  moments,  is  — - —  X  —  X  —  = •  — . 

/          Jj        ct         Z  d      I 

Comparing  this  with  the  former  result,  we  find  that  the 
horizontal  strain  at  8  is  to  the  force  that  would  simply  sustain 
the  part  S  0  as  (/  —  u)  is  to  u ;  and  consequently,  it  is  only 

when  u  —  —  or  the  point  $  coincides  with  the  centre,  that  the 

horizontal  strain  is  equal  to  the  force  that  would  sustain  either 
portion  in  equilibrio,  if  the  beam  be  supposed  to  be  cut  through 
at  that  point. 

As  (/  —  u)  is  greater  than  u  for  any  point  between  0  and 
the  centre,  it  follows,  that  the  horizontal  strain  is  greater  than 
would  be  produced  simply  by  the  pressure  of  the  weight  on  S 
0  if  free  to  move  around  B. 

If  we  make  u  =  o,  both  the  above  strains  become  o,  but 
the  proportion  /  —  u  :  u  would  give  /  :  o.  This  result  is  a 
consequence  of  the  omission  of  the  factor  u  which  is  common 


WOODEN   BRIDGES.  69 

to  both  terms  ;  if  it  be  introduced,  the  proportion  becomes  u 
(I  —  u}  :  U2y  which  reduces  to  o  :  o  when  u  =  o  and  involves 
no  contradiction. 

w  u^ 

The  strain  at  any  point  being  ^  (u  —  -y)  will  be  a  maxi- 

u2 
mum  whenever  (11  --  ,-)  is  a  maximum.     The  first  differen- 

[f 

tial  coefficient  of  this  expression  being  placed  =  o,  we  have 
1  —  ---  =  o,  or  u  =  -Q.  Hence,  the  maximum  strain  is  in  the 

centre. 

the  force  necessary  to  sustain  the  part  S  D,  if  applied  at 

S,  in  the  direction  S  Z>,  is  ^-\  (  —  j  —  )  . 

Comparing  this  with  the  actual  strain  at  $,  which  is  ^-3 

—  -  —  )  u,  we  find  the  former  to  be  the  greater  in  the  proportion 

of  (I  —  u)  :  u. 

Consequences  which  are  deduced  from  the  above  results. 

1.  Since  the  horizontal  strain  at  the  centre  is  exactly  equal 
to  that  which  would  sustain  one-half  of  the  beam,  if  the  other 
half  should  be  removed,  it  follows,  that  there  can  be  no  verti 
cal  strain  at  this  point. 

2.  As  the  actual  horizontal  strain  at  any  other  point  S  is 
less  than  the  force  that  would  sustain  the  part  S  J)  in  the  pro 
portion  of  u  to  /  —  U,  it  follows,  that  the  part  S  D  must  press 
vertically  and  produce  a  vertical  strain,  which  would  be  mea 
sured  by  a  vertical  force  sufficient  to  compensate  for  the  diffe 
rence  of  the  horizontal  forces. 

As  the  horizontal  forces  are  proportional  to  u  and  (I  —  u). 
If  H  represent  the  strain  at  $,  we  have  u  :  I  —  u  :  :  H  : 

H  (  -rr—  )i  and  the  difference  of  the  forces  will  be  II 


This  force  is  not  resisted  by  any  antagonist  horizontal  force, 
and  must  therefore  produce  a  vertical  strain  on  the  fibres,  the 


70  BRIDGE   CONSTRUCTION. 

measure  of  which  would  be  the  force  which  acting  vertically 
would  give  the  same  moment  in  reference  to  the  point  A. 
Call  wf  this  force;  its  leverage  is  (I  —  u).     The  moments 

will  be  U  \~~        ~)  d  and  wf  (I  —  w),  by  which  we  obtain 
=  H  d  (—TTI.  —\)«     Substitute  the  value  of  H  =  ^ 


w 

(  —  j  —  )  we  have  w'  =  ~~,  (I  —  2  u)  =  The  expression  for 

the  vertical  strain  at  any  point  S  which  becomes  zero  at  the 

w 
centre,  or  when  u  =  J  1.     When  u  =  o  we  have  w'  —  -?  = 

strain  at  the  ends. 

w 

This  expression  nj(l  —  2  u)  is  equal  to  the  difference 

S  of  —  S  o"  of  the  vertical  components  for 


l:u::$w:  TT7  = 

w  (I  —  u       w  u       w 


Hence  o'  o"  represents  the  vertical  strain  at  any  point. 

w  w  w 

The  expression  ~-,  (I  —  2  u)  =  —  -=  u  +  -^  bring  of  the 

form,  y  =  —  a  x  -\-  b  is  the  expression  for  a  straight  line,  and 
therefore  if  A  B  represent  a  beam,  and  B  n  =  J  w  the  straight 
line,  c  n  will  determine  the  perpendiculars  p  m  which  measure 
the  strains. 


FIG.  39. 


To  show  this  more  clearly  let  u  =  ~  —  a;  whence  x  =  ^ 


WOODEN    BRIDGES.  71 

—  u  =  c  p.     Substitute  the  value  of  u  and  reduce:  the  ex- 

w          w  ,  w 

pression  —  y  M  +  9"  becomes  y  x  which  is  the  equation   of 

a  straight  line  passing  through  the  origin  C. 

'  It  follows  therefore  that  the  vertical  strains  are  exactly  pro 
portional  to  the  distance  from  the  centre,  a  consequence  of  the 
greatest  importance  in  its  application  to  the  practice  of  Bridge 
Construction. 

To  find  the  curve  which  represents  the  horizontal  strain. 
The  horizontal  strain  at  any  point  of  a  beam  supported  at 

the  ends  and  loaded  uniformly,  was  found  to  be   ^  ,    X  — j — 

in  which  u  represents  the  distance  of  the  point  from  the  end, 
w  =  the  whole  weight,  I  =  the  length,  and  d  =  the  depth  of 

NOTE. — It  may  be  thought  that  the  principle  which  I  have  endeavored 
to  establish  is  too  simple  to  require  the  explanation  that  has  been  given  ; 
but  simple  as  it  is,  the  consequences  are  important,  and  I  do  not  know 
that  it  has  been  noticed  by  writers  upon  Bridge  Construction  or  the  re 
sistance  of  Solids  ;  certain  it  is  that  the  effects  which  naturally  result  from 
it  have  been  overlooked  in  proportioning  structures.  In  fact  it  was  not 
until  some  months  after  my  attention  had  been  directed  to  the  theory  of 
Bridge  Construction,  that  I  was  led  to  observe  the  difference  in  the  vertical 
forces  at  different  points  of  a  straight  truss.  The  fact  that  such  difference 
exists  was  first  pointed  out  to  me  by  II.  R.  Campbell,  of  Phila.,  a  gentle 
man  who  in  the  course  of  a  long  and  extensive  practice  as  a  Civil  Engi 
neer,  has  enjoyed  rare  opportunities  for  becoming  acquainted  with  Bridgo 
Construction  and  for  observing  the  effects  of  time  and  accidents. 

In  a  conversation  with  him  upon  the  principles  of  the  art,  he  asked  me 
to  explain  \vhy  the  chords  of  a  Bridge  which  had  settled  considerably 
were  more  bent  at  the  abutments  than  at  the  middle.  I  had  not  then 
particularly  noticed  the  fact,  but  he  assured  me  that  although  the  de 
pression  was  greatest  in  the  middle  when  a  straight  Bridge  settled  below 
its  level,  yet  the  curvature  was  not  uniform,  and  the  quickest  bend,  or 
in  other  words  the  least  radius  of  curvature,  was  always  nearest  the 
abutment.  lu  a  subsequent  examination  of  a  large  number  of  bridges, 
I  invariably  found  that  the  joints  of  the  braces  near  the  abutments  were 
compressed  and  tight,  whilst  near  the  centre  of  the  spans  no  symptoms 
of  crushing  were  perceptible,  and  in  some  cases  where  the  joints  of  the 
central  braces  were  not  well  fitted,  a  knife  blade  could  be  introduced, 
clearly  indicating  a  great  increase  of  pressure  towards  the  abutments, 
and  as  a  consequence,  the  necessity  of  increasing  the  number  or  size  of 
the  vertical  supports  towards  the  extremities. — Author. 


72  BRIDGE   CONSTRUCTION. 

the   beam.      This   expression    can   be   put   under   the   form 
w  w 

— — -  u u  . 

FIG.  40. 


If  we  make  u  =  o  or  u  =  I  the  expression  in  either  case 
becomes  0,  and  if  we  express  the  values  of  the  strains  by  the 
ordinates  of  a  curve  of  which  the  above  is  the  equation,  we 
find  that  the  curve  passes  through  the  points  B  and  A  at  the 
ends  of  the  beam. 

w  w 

If  we    differentiate   the   expression  ^~T  u  —  o~T/  u  2   an(^ 

place  the  first  differential  co-efficient  equal  to  zero  :  we  have 

w  w  I 

cTj  —  cTTT  u  ~  °  whence  u  =.5  and  the  maximum  is  at  the 

J  d        A  cL  L  £ 

centre  Q. 

The  value  of  the  maximum    strain  found  by  substituting 

/  wl 

5  for  u  is  g-y.     Let  this  value  be  represented  by  the  line  (7, 

—  o  CL 

p  and  p  will  be  a  point  of  the  curve. 

To  ascertain  the  nature  of  the  curve,  we  will  transfer  the 
origin  from  B  (the  point  from  which  u  is  reckoned)  to  p. 

First  make  u  =  (^  —  x)  we  obtain  for  the  value   of  the 

w  I          w 
ordinate  when  the  expression  is  reduced  y  =  g-j  —  9^7  x* 

which  is  the  equation  of  the  curve  when  the  origin  is  at  0. 
Agair*   make  y  —  ^-j  —  yr    and   we    obtain  y'  =  ^rry  x2, 


which  is  the  equation  of  the  curve  when  the  origin  is 
but  this  equation  is  that  of  a  parabola  ;  hence,  the  ordinate  of 
a  parabola  drawn  through  B,p,  and  A  will  exhibit  the  inten 
sity  of  the  horizontal  strain  at  any  point,  and  furnishes  a  geo 
metrical  method  of  obtaining  it. 


WOODEN  BRIDGES. 


73 


To  find  the  pressure  upon  the  supports  when  a  beam  is 
framed  as  a  cap  upon  the  tops  of  several  vertical  posts,  and 
a  weight  applied  directly  over  one  of  the  2^osts. 

This  is  a  case  which  may  be  of  use  in  proportioning  the 
timbers  for  bridges  when  the  supports  are  close  together. 


FIG.  41. 

c 

Al 

If  we  suppose  the  material  of  the  posts  to  be  perfectly  in 
elastic,  the  middle  one  would  bear  the  whole  of  a  weight  ap 
plied  at  0  and  no  part  of  it  would  be  sustained  by  A  and  B  : 
but  if  the  beam  be  flexible  and  the  substance  of  the  posts 
clastic,  the  pressures  upon  A  and  B  would  depend  upon  the 
relative  degrees  to  which  these  properties  were  possessed.  If 
the  beam  be  very  stiff  and  the  posts  elastic,  a  large  part  of  the 
pressure  will  be  thrown  upon  A  and  B,  and  if  the  beam  be 
very  flexible  and  the  post  but  slightly  elastic,  nearly  all  the 
weight  will  be  sustained  at  Q. 

When  the  distance  between  the  supports  and  the  dimen 
sions  of  a  beam  are  known,  the  flexure  caused  by  a  given 
weight  is  readily  calculated :  and  when  the  length  of  a  sup 
port  is  known,  the  reduction  in  length  due  to  a  given  weight 
can  also  be  determined. 

If  w  represent  the  weight  at  (7,  d  =  the  deflection  which 
would  be  produced  if  the  support  were  removed,  e  =  the  re 
duction  in  length  by  the  same  weight  which  the  post  would 
experience.  Then  if  x  represent  the  actual  deflection,  we  will 
have,  since  the  deflection  is  always  proportional  to  the  weight, 

=  weight    sustained    by.  the   beam    and 


W  X 


w 


which  is  transmitted  to  the  points  A  and  B. 


Also,  e  :  x  : :  w  : 


w  x 


=  weight  sustained  by  post  Oy  the 


74  BRIDGE   CONSTRUCTION. 

sum  of  these  "weights  must  be  equal  to  w.     We  therefore  have 

w  x        w  x  /         -7\  i  e  d 

4-  =  w  or  (e  +  d)  x  =  e  d,  x  — 

d  e  e  +  d 

Substituting  this  value  we  find 

For  the  pressure  upon  A  and  B  =  ~L  —  —^e 

_,       ,  „       w  x        w  d 

For  the  pressure  upon  C  ==• = - 

e         I  +  d 

If  the  ends  of  the  posts  instead  of  resting  against  solid  point3 
of  support,  be  placed  upon  a  second  beam,  the  circumstances 
of  the  case  will  be  very  different. 

FIG.  42. 


Let  A  0  and  D  F  be  two  equal  beams  connected  by  an 
upright  in  the  centre  and  loaded  with  a  weight  at  B. 

If  we  suppose  B  E  to  be  perfectly  incompressible,  then  in 
case  of  flexure  a  c  and  D  F  would  retain  their  parallel  posi 
tions,  and  each  would  assist  equally  in  sustaining  the  load,  the 
post  would  then  be  pressed  upwards  against  the  point  B  with 

*M 

a  force  equal  to  the  reaction  of  the  lower  beam  or  equal  to  — . 
But  if  the  post  be  elastic  it  will  be  compressed  to  some  ex- 

IV 

tent  by  the  action  of  — ,  and  as  a  consequence,  D  F  would 

rise,  and  the  deflection  becoming  less  it  would  sustain  less  of 
the  weight.  A  0  must  then  sink  lower  to  compensate  for  this 
diminished  strain  on  the  lower  beam,  and  in  proportion  to  the 
elasticity  of  B  e  will  be  the  difference  of  the  strains  upon  A  c 
and  D  F. 

To  determine  the  strains  and  deflections  of  the  beams  and 
the  degree  of  compression  of  the  posts  by  calculation.  Let  the 
beams  bo  supposed  of  any  relative  size,  and  to  make  the  case 


WOODEN   BRIDGES.  75 

general,  let  the  stiffness  of  the  lower  be  to  that  of  the  upper 
as  n  :  1. 

Also  let  w  =  weight  at  B,  d  =  the  deflection  that  it  would 
produce  in  the  distance  A  c,  e  =  the  compression  of  the  post 
by  the  same  weight,  x  =  the  actual  deflection  in  the  upper 

w 

beam.     Then  d  :  x  :  :  10  :  -7  x  =  weight  sustained  by  upper 

ct 

10  % 

beam,  and  w  —  -7  x  =  w  (1  —  -r)  =  portion  of  weight  trans 
mitted  to  lower  beam. 

The  deflection  of  the  lower  beam  by  the  weight  w  is  n  d, 
hence  the  actual  deflection  will  be  determined  by  the  propor- 

/7*  ^7* 

tion  w  :  n  d  :  :  w  (1  —  -j)  :  n  d  (1  —  -5 )  =  n  d  —  n  x  = 

n  (d  —  x). 

The  difference  between  the  deflections  of  the  beams  must 
give  the  compression  of  the  post,  which  is  accordingly  equal 
to  x  —  n  (d  —  x)  =  (n  +  1)  x  —  n  d.  But  the  compression 
of  the  post  as  determined  from  the  pressure  will  be  w  :  e  :  : 

/>»  /-v* 

w  (1 —  -T)  :  e  (1 A  equating  these  results  we  have  e  — 

ex  d  (n  d  +  e) 

—-  —  (n  +  1)  x  —  n  d,  whence  x  =  -, ,   1X   ,    ,     .       This 

d  (n  +  1)  d  +  e 

w                               x 
value   substituted   in   the    expressions  -7    x   and  w  (1 j) 

will  give  the  portions  of  the  weight  sustained  by  the  upper 
and  lower  beams,  and  by  the  post. 

From  the  above  we  learn  that  when  the  beams  are  equal, 
the  pressure  upon  the  post  is  always  less  than  J  W. 

For  in  this  case  n  —  1  and  x  =  -?TT~, — ~  when  e  =  o  or 

2  d  +  e 

d2         d 
the  post  is  incompressible,  we  have  x  —  ^-,  =  -^-,  and  each 

beam  bears  half  the  weight,  consequently  the  strain  upon  the 
post,  which  is  always  equal  to  that  upon  the  lower  beam,  will 
be  J  W.  If  e  be  not  o  the  value  of  x  will  be  greater  than 

•jr-,  and  consequently  the  post  will  transmit  to  the  lower  beam 
22 

W 

less  than     -. 


}6  BRIDGE   CONSTRUCTION. 

The  pressure  upon  the  points  A  and  0  will  be  each  one 
naif  of  the  weight  sustained  by  the  upper  beam,  and  on  the 
points  D  and  F  one-half  of  the  weight  on  the  lower  beam. 


Strength  of  a  long  beam  laid  over  several  supports. 

This  subject  properly  belongs  to  the  resistance  of  timber ; 
but  as  it  expresses  so  nearly  the  condition  of  a  continuous 
bridge  supported  by  a  number  of  piers,  it  has  been  considered 
preferable  to  introduce  it  in  this  place. 

FIG.  43. 

m  P          TZ  f  n 


k  -  B  -  [  -  z  -  >  -  1 
A.  Jl  CO) 

Let  A  B  0  D  represent  a  beam  laid  over  several  supports, 
and  loaded  with  a  uniform  weight.  If  we  examine  the  cen 
tral  interval,  we  perceive  that  the  weight  upon  it  is  sustained 
by  the  resistance  of  the  sections  at  m,  p,  and  n,  and  the  whole 
weight  would  be  equal  to  the  sum  of  the  weights  that  each 
section  separately  would  be  capable  of  sustaining. 

The  resistance  of  each  section  being  R  d2.  If  w  represent 
the  uniform  weight  upon  the  whole  beam,  we  will  have  for  the 
weight  that  the  section  m  alone  could  sustain 

Ed2 

R  d2  =  J  w  x  J  I  =  or        w  =      4  —  j— 


t  Rd2 
For  the  section  at  n  w  =      4  —j— 

T)     -J2 

For  the  middle  section         w  =      8  —  -,  — 


Rd2 

And  for  the  whole  weight  16  — ^ — • 

which  ib  twice  the  weight  that  the  middle  section  alone  ia 
capable  of  sustaining,  or  in  other  words :  The  strength  of  a 
beam  fixed  at  the  ends  is  to  the  strength  of  a  beam  free  at  the 
ends  as  2  is  to  1. 

For  the  end  section  (n  o)  we  have  weight  which  the  sec- 


"WOODEN   BRIDGES.  77 


4 

tion  at  n  alone  would  sustain  to  ==  4  —  j  — 

Weight  which  the  section  at  o  would  sustain          w  =      0 

Rd2 

Weight  which  the  section  at  pr  would  sustain        w  —  8  —  — 


Total 

The  strength  of  the  end  span,  or  of  a  beam  fixed  at  one 
end  and  free  at  the  other,  is  to  the  strength  of  a  beam  free  at 
both  ends  as  12  :  8,  or  as  3  :  2. 

When  the  span  becomes  considerable,  simple  timbers  are 
insufficient,  and  framed  trusses  become  necessary.  Whatever 
may  be  their  particular  form,  the  object  in  every  case  obviously 
is,  to  dispose  of  a  given  quantity  of  material  so  as  to  resist  ef 
fectually  all  the  forces  which  tend  to  produce  rupture  or  change 
of  form. 

The  consideration  of  the  case  of  a  single  beam  involves  the 
principles  of  a  framed  truss,  the  same  forces  act  in  both,  the 
manner  of  resisting  them  alone  is  different:  in  the  former,  the 
cohesion  of  the  fibres  secures  the  object  ;  in  the  latter,  it  must 
be  attained  by  a  judicious  combination  of  ties  and  braces. 

It  has  been  shown,  that  in  a  beam  the  parts  near  the  axis 
are  but  little  strained,  and  consequently  oppose  but  little  resist- 
tance  ;  hence,  they  could  be  removed  without  serious  injury  ; 
and,  if  the  same  amount  of  material  could  be  disposed  at  a 
greater  distance  from  the  axis,  the  strength  and  stiffness  would 
be  increased  in  exact  proportion  to  the  distance  at  which  they 
could  be  made  to  act  :  hence,  the  first  object  in  designing  a 
truss,  must  be,  to  place  the  material  to  resist  the  horizontal 
forces  at  the  greatest  distance  from  the  neutral  axis,  which 
the  nature  of  the  structure  will  allow. 

It  is  evident,  however,  that  if  two  longitudinal  timbers 
should  be  placed  parallel  to  each  other,  without  intermediate 
connections,  nothing  would  be  gained  ;  for,  in  this  case,  each 
would  act  independently  of  the  other,  and  the  strength  would 
be  less  than  that  of  a  single  beam.  Neither  would  a  connection 
by  means  of  vertical  ties,  as  in  the  figure,  add  to  the  strength  ; 
for,  the  weight  of  the  ties  would  increase  the  load,  to  resist 


78 


BRIDGE   CONSTRUCTION. 


FlG.  44, 


which,  there  would  be  only  the  stiffness  and  strength  of  the 
beams  A  B  and  CD. 

By  observing  the  effect  of  flexure  upon  this  system,  we  are 
at  once  enabled  to  perceive  the  means  by  which  it  can  be  pre 
vented. 

The  rectangles  formed  by  the  horizontal  and  vertical  pieces 
are  converted  into  oblique  angled  parallelograms,  one  diagonal 
of  the  rectangle,  as  A  m,  being  lengthened,  and  the  other,  as 
On,  shortened;  and,  as  this  effect  must  take  place  to  a  greater 
or  less  extent  whenever  any  degree  of  flexure  is  produced,  it 
may  be  concluded,  that  the  introduction  of  braces  which  would 
prevent  any  change  of  figure  in  the  rectangles  will  effectually 
prevent  flexure.  This  is  in  fact  the  case,  and  the  combination 
of  timbers  represented  in  figure  45  is  sufficient  to  form  a  com 
plete  truss,  capable,  when  properly  proportioned,  of  resisting  the 
action  of  any  uniform  load. 

FIG.  45. 


It  appears,  therefore,  that  in  the  construction  of  the  vertical 
frame  or  truss  of  a  bridge,  at  least,  three  series  of  timbers  enter 
as  indispensable  elements ;  these  may  be  called,  chords,  ties, 
and  braces,  and  these  are  all  that  any  uniform  load  requires. 

The  manner  in  which  such  a  combination  of  parts  acts  to 
sustain  a  weight  will  now  be  examined. 

CASE  1.  Let  the  weight  be  uniformly  distributed  upon 
A  B.  It  is  evident  that  in  case  of  flexure  the  depression  will 
be  greatest  in  the  middle. 


WOODEN"    BRIDGES. 


79 


All  the  diagonals  of  the  rectangles,  in  the  direction  of  the 
braces,  will  have  a  tendency  to  shorten  ;  and,  as  this  is  effectu 
ally  resisted  by  the  braces,  it  follows,  that  such  a  truss  is  fully 
capable  of  sustaining  a  weight  thus  distributed. 

CASE  2.  Let  the  weight,  instead  of  being  uniformly  dis 
tributed  along  B  E,  be  applied  at  some  point  Cf. 


If  we  represent  it  by  the  portion  Cf  p  of  its  line  of  direction, 
and  construct  the  parallelogram  of  forces  on  C'  0  and  Cr  D, 
we  find  0'  o  =  w  cosec.  a  —  strain  on  0'  D. 

If  the  point  of  application  be  removed  to  D,  and  again  re 
solved  into  vertical  and  horizontal  components,  the  vertical 
force  will  be  equal  to  D  pf  =  w.  But  this  result  is  evidently 
false,  for  the  weight  is  sustained  by  the  points  A  and  D,  and 
presses  upon  them  in  proportion  to  the  distances  C'  B  and  C' JS9 
it  cannot  therefore  be  equal  to  w  at  either.  As  cases  of  this 
kind  frequently  occur  in  attempting  to  trace  the  effects  offerees 
upon  the  parts  of  a  connected  system,  and  often  lead  to  error, 
we  will  endeavor  to  explain  the  cause  of  this  apparently  para 
doxical  result,  which  seems  to  contradict  established  principles. 

If  we  suppose  two  inflexible  rods,  one  horizontal  and  the 
other  vertical,  to  be  loaded  with  a  weight  applied  at  the  angu 
lar  point,  A  and  D  both  resting  against  fixed  points,  then,  the 
weight  being  represented  by  the  portion  Op  of  its  line  of  direc 
tion,  maybe  resolve  1  into  components  in  the  dire-ction  of  0  A 
and  CD. 

FIG.  47. 


80 


BRIDGE   CONSTKUCTION. 


The  horizontal  component  can  produce  no  vertical  actior 
at  the  point  A,  but,  the  oblique  component,  being  transferred  to 
the  fixed  point  D  and  resolved  into  horizontal  and  vertical 
forces,  will  give  a  vertical  pressure  equal  to  Op.  This  result 
is  correct,  because  A  sustains  no  vertical  pressure  and  conse 
quently  bears  no  portion  of  the  weight,  which  must,  therefore, 
be  thrown  entirely  upon  D. 

This  case,  however,  does  not  present  the  true  condition  of 
the  problem,  for  the  upper  chord  does  not  abut  against  a  fixed 
point  at  its  extremity ;  if  it  did,  the  result  would  be  correct,  for 
it  is  evident,  that  a  sufficient  force  applied  at  B,  in  the  direction 
B  (7,  would  raise  the  truss,  causing  it  to  revolve  around  the 
point  D,  and  relieving  A  of  any  portion  of  the  pressure. 

FIG.  48. 


In  consequence  of  the  connection  of  the  parts  of  the  frame, 
the  true  line  of  direction  will  be  0  A,  and  the  strains  must  bo 
estimated  by  resolving  .the  weight  into  components  in  the  di 
rections  0 A  and  QD. 

FIG.  49. 


To  simplify  the  demonstration,  let  the  lines  A  0  and  0  D 
be  loaded  at  (7,  and  the  weight  represented  by  Onf  resolved 
into  components ;  On  will  represent  the  strain  on  CD,  and  n 
nf  —  the  strain  on  A  0.  By  transferring  the  force  On  to  the 
point  D,  and  resolving  it  into  horizontal  and  vertical  compo 
nents,  we  find  the  pressure  on  D  to  be  equal  to  Co;  and,  in 
like  manner,  the  pressure  on  A  will  be  found  to  be  o  nr.  But, 
from  similar  triangles,  we  have 


WOODEN   BRIDGES. 


81 


0  o  :  Op  :  :  on  :  p  D 
and     Op  :  o  n'  :  :  A  p  :  on 

Hence,  Oo  :  o  nr  :  :  Ap  :  p  D  a  result,  which, 
as  it  makes  the  pressure  upon  A  and  D  proportional  to  their 
distances  from  the  line  of  application  of  the  weight,  must  be 
correct. 

We  have  seen,  that  in  estimating  the  effect  of  a  weight  at 
0')  it  is  necessary  to  resolve  it  into  components  in  the  directions 
A  Of  and  B  G'. 

In  the  same  manner,  it  can  be  shown  that  the  forces  which 
act  at  0  must  by  the  connection  of  the  system  be  transferred 
to  the  points  B  and  A,  in  the  directions  of  the  diagonals  A  0 
and  B  0. 


FIG.  50. 


c' 


The  effect  of  the  oblique  force  Cf  A  upon  the  angle  (7  evi 
dently  is  to  force  it  upwards,  and  the  strain  would  be  the  dia 
gonal  of  a  parallelogram  constructed  upon  A  0  and  0  0'. 

This  result  is  of  the  greatest  practical  importance,  and  the 
existence  of  a  force  acting  upwards  appears  to  have  been  over 
looked  by  many  practical  builders,  as  in  some  very  important 
structures  no  means  have  been  used  to  guard  against  its  effects. 

The  consequence  is,  that  in  a  straight  as  well  as  in  an 
arched  truss,  a  weight  at  one  side  produces  a  tendency  to  rise, 
at  the  other  side. 

FIG.  51. 


The  effect  of  this  upward  force  is  to  compress  the  diagonals 
in  the  direction  of  the  dotted  lines  and  extend  them  in  the  di 
rection  of  the  braces ;  but  as  the  braces,  from  the  manner  in 
which  they  are  usually  connected  with  the  frame,  are  not  capa* 
6 


82  BRIDGE    CONSTRUCTION. 

ble  of  opposing  any  force  of  extension,  it  follows,  that  the  only 
resistance  is  that  which  is  due  to  the  weight  and  inertia  of  a 
part  of  the  structure.  When  the  load  is  uniform  this  is  suffi 
cient,  because  the  weight  on  one  side  is  balanced  by  an  equal 
weight  upon  the  other,  and  every  part  is  in  equilibrium.  But, 
when  the  bridge  is  subjected  to  the  action  of  a  heavy  weight, 
as  a  locomotive  engine  or  a  loaded  car  rapidly  passing  over  it, 
and  acting  with  impulsive  energy  upon  every  part  at  different 
instants,  it  is  obvious,  that  no  adequate  resistance  is  offered  by 
a  truss  composed  of  only  the  three  series  of  timbers  already  de 
scribed.  Yet,  we  find  that  such  a  truss  has  been  used  for  a 
large  proportion  of  the  bridges  that  have  been  erected,  some 
times  with,  and  sometimes  without  the  addition  of  an  arch,  an 
appendage  which,  although  it  adds  to  the  vertical  strength,  di 
minishes  but  little  the  effect  of  the  force  under  consideration. 
No  one,  who  has  had  an  opportunity  of  observing  it,  can  have 
failed  to  notice  the  great  vibration  produced  in  such  bridges  by 
the  passage  of  a  loaded  vehicle.  In  long  bridges,  the  undula 
tions  produced  by  the  passage  of  a  car  can  be  felt  at  the  dis 
tance  of  several  spans. 

The  remedy  for  this  defect  is  obvious  ;  it  is  only  necessary 
to  prevent  the  diagonal,  in  the  direction  of  the  dotted  line,  from 
shortening,  or,  in  the  direction  of  the  brace,  from  lengthening, 
and  the  upward  force  will  be  effectually  resisted. 

This  requires,  either,  that  counter-braces  should  be  intro 
duced  in  the  directions  of  the  dotted  diagonals  of  the  last  figure, 
or,  that  the  braces  themselves  should  be  capable  of  acting  as 
ties,  or,  additional  ties  placed  in  the  direction  of  the  braces. 

It  follows,  from  the  preceding  exhibition  of  the  effect  of  a 
variable  load,  that  no  bridge,  either  straight  or  arched,  which 
is  designed  for  the  passage  of  vehicles,  and  particularly  of  rail 
road  trains,  should  be  constructed  without  counter-bracing  or 
diagonal  ties.  It  is  only  in  aqueducts,  where  the  load  is 
always  uniform,  that  they  can  with  any  propriety  be  omitted. 

Effects  of  counter-bracing. 

The  consideration  of  the  action  of  counter-braces  leads  to 
some  very  singular  but  important  results. 


TTOODEN   BRIDGES.  83 

Let  the  truss  be  loaded  with  a  weight  so  as  to  produce 
some  deflection,  it  has  been  shown  that  the  diagonals  in  the 
direction  of  the  braces  will  be  compressed,  and  in  the  direction 
of  the  counter-braces  extended.  Suppose  that  the  extension  of 
the  last  named  diagonals  is  sufficient  to  leave  an  appreciable 
interval  between  the  end  of  the  counter-brace  and  the  joint 
against  which  it  abuts,  and  that  into  this  interval  a  key,  or 
wedge  of  hard  wood  or  iron,  is  tightly  introduced :  it  is  evident, 
that  upon  the  removal  of  the  weight,  the  truss,  by  virtue  of  its 
elasticity,  would  tend  to  regain  its  original  position ;  but  this  it 
cannot  do,  in  consequence  of  the  wedges  at  the  ends  of  the 
counter-braces,  which  prevent  the  dotted  diagonals  from  reco 
vering  their  original  length,  and  the  truss  is  therefore  forcibly 
held  in  the  position  in  which  the  weight  left  it ;  the  reaction  of 
the  counter-braces  producing  the  same  effect  that  was  produced 
by  the  weight,  and  continuing  the  same  strain  upon  the  ties 
and  braces. 

The  singular  consequence  necessarily  results  from  this,  that 
the  passage  of  a  load  produces  no  additional  strain  upon  any 
of  the  timbers,  but  actually  leaves  some  of  them  without  any 
strain  at  all. 

FIG.  52. 


To  render  the  truth  of  this  assertion  more  clear,  we  will 
confine  ourselves  to  the  consideration  of  a  single  rectangle,  and 
suppose  that  the  effect  of  the  flexure  caused  by  an  applied 
weight  has  been  to  extend  the  diagonal  A  0  by  a  length  equal 
to  A  j9,  and  to  compress  the  brace  B  D  by  an  equal  amount. 

The  point  p  will  evidently  be  drawn  away  from  J.,  leaving 
the  interval  A  p.  If  a  wedge  be  tightly  fitte  1  into  this  interval 
without  being  forcibly  driven,  it  evidently  can  have  no  action 


84  BEIDGE   CONSTRUCTION. 

upon  the  frame  so  long  as  the  weight  continues  ;  but  upon  the 
removal  of  th.e  weight  it  becomes  forcibly  compressed,  in  con 
sequence  of  the  effort  of  the  truss,  by  virtue  of  its  elasticity,  to 
return  to  its  former  position.  This  effort  is  resisted  by  the  reac 
tion  of  the  wedge,  which  causes  a  strain  upon  the  counter-brace 
A  C  sufficient  to  counteract  the  elasticity  of  the  truss ;  and,  as 
no  change  of  figure  can  take  place,  it  follows,  that  the  brace  B 
D  cannot  recover  its  original  length,  and,  therefore,  continues 
as  much  compressed  as  it  was  by  the  action  of  the  weight. 

The  effect  of  a  weight  equal  to  that  first  applied  will  be  to 
relieve  the  counter-brace  A  (7,  without  adding  in  the  slightest 
degree  to  the  strain  upon  B  D. 

As  regards  the  effects  upon  the  chords,  it  is  evident  that  the 
strains  are  only  partial,  and  tend  to  counteract  each  other. 
The  maximum  strain  in  the  centre  is  estimated  by  the  force 
which  would  be  required  to  hold  the  half  truss  in  equilibrium 
if  the  other  half  be  removed ;  and  this  is  dependent  only  on 
the  weight  and  dimensions  of  the  truss.  In  fact,  if  we  ex 
amine  the  parallelogram  A  B  CD,  we  find  that  the  effect  of 
wedging  the  diagonals  will  be  to  produce  strains  acting  in 
opposite  directions  at  A  and  B,  and  destroying  each  other's 
effects ;  the  strains  produced  by  wedging  any  rectangle  cannot 
therefore  be  continued  to  the  next,  and  of  course  can  have  no 
influence  upon  the  maximum  forces  at  the  centre. 

As  the  vibration  of  a  bridge  is  caused  principally  by  the 
effort  to  recover  its  original  figure  after  the  compression  pro 
duced  by  a  passing  load,  it  follows,  that  if  this  effort  is  resisted, 
the  vibration  must  be  greatly  diminished,  or  almost  entirely 
destroyed. 

This  accounts  for  the  surprising  stiffness  which  is  found  to 
result  from  a  well-arranged  system  of  counter-braces. 

Inclination  of  braces. 

From  the  preceding  examination  into  the  distribution  of  the 
forces,  we  learn  that  at  least  four  sets  of  timbers  are  necessary 
in  every  complete  and  well-arranged  truss. 

The  proper  disposition,  and  the  relative  proportion  of  the 
parts,  next  demand  attention. 


WOODEN   BRIDGES.  85 

The  horizontal  strain  in  the  centre  of  the  span,  being  equal 
to  the  force  which  would  sustain  the  half  truss  in  equilibrium, 
is  independent  of  the  particular  number  or  inclination  of  the 
braces.  The  same  may  be  said  of  the  pressure  upon  the  abut 
ments,  which  is  always  proportional  to  the  distance  of  the  cen 
tre  of  gravity  from  the  point  of  support  at  the  other  end  of  the 
truss. 

The  parts  of  a  frame  can  only  act  by  distributing  the  forces 
which  are  applied  to  it, — they  cannot  create  force  ;  hence  what 
ever  be  the  inclination  of  the  braces,  the  pressure  upon  the 
abutment  and  the  strain  upon  the  centre  of  the  chord  must 
remain  the  same,  with  the  same  weight.  It  might  be  inferred, 
therefore,  that  the  degree  of  inclination  was  of  little  conse 
quence,  or  that  different  angles  would  be  equally  advantageous. 
That  such  is  not  the  case  can  be  rendered  evident  by  the  fol 
lowing  considerations. 

1.  The  braces  must  not  be  so  long  as  to  yield  by  lateral 
flexure. 

2.  The  chords  being  unsupported  in  the  intervals  between 
the  ties,  these  intervals  must  be  limited  by  the  condition  that 
no  injurious  flexure  shall  be  produced  by  the  passage  of  a  load. 

On  the  other  hand,  as  the  ties  approach  each  other,  the 
angle  of  the  brace  increases ;  and  when  the  intervals  become 
small,  the  number  of  ties  and  braces  is  greatly  increased,  and 
with  them  the  weight  of  the  structure. 

The  true  limit  of  the  intervals  can  be  readily  determined 
when  the  size  of  the  chords  and  the  maximum  load  are  known  ; 
for  it  should  evidently  be  such  that  when  the  load  is  at  the 
middle,  the  flexure  should  not  exceed  a  given  amount. 

The  portion  of  the  chord  between  any  two  ties  is  in  the 
condition  of  a  beam  supported  at  the  ends  and  loaded  in  the 
middle. 

Should  the  angle  of  the  brace  as  determined  by  this  con 
dition  be  too  great,  the  remedy  consists  in  introducing  inter 
mediate  timbers  as  represented  by  the  dotted  lines  in  the  mar 
ginal  figure,  and  it  is  evident  that  by  the  addition  of  these  we 
are  enabled  to  vary  the  inclination  at  pleasure.  A  system  of 
framing  that  will  admit  of  the  introduction  of  such  timber? 


86 


BRIDGE    CONSTRUCTION. 


may  therefore  prove  very  advantageous,*  provided  they  are 
not  introduced  at  the  sacrifice  of  counter-braces  or  diagonal 
ties. 

FIG.  53. 


To  determine  the  strain  upon  the  counter-braces. 

It  is  evident  in  the  first  place  that  counter-braces  do  not 
assist  in  sustaining  the  weight  of  the  structure;  on  the  contrary, 
the  greater  the  weight  and  the  degree  of  flexure  which  it  oc 
casions,  the  more  will  the  counter-braces  be  relieved. 

The  strain  under  consideration  must  therefore  result  from 
the  variable  load,  and  as  the  effect  of  a  weight  on  one  side  of  a 
truss  is  to  produce  a  tendency  to  rise  at  the  opposite  side,  which 
is  resisted  by  the  counter-braces  ;  and  as  this  strain  is  not  con 
fined  to  a  single  timber,  but  distributed  amongst  several ;  as  it 
is  also  resisted  by  the  weight  of  portions  of  the  structure,  mod 
ified  by  the  nature  of  the  connections  and  the  degree  of  flexure 
which  the  timbers  experience,  it  would  be  a  very  complicated 
problem  to  trace  the  effects  of  a  weight  through  the  system  of 
timbers  which  compose  the  truss,  in  order  to  determine  the 
maximum  strain  upon  a  single  timber.  Fortunately  we  have 
a  more  direct  and  accurate  means  of  obtaining  the  result. 

It  has  been  shown  that  by  driving  a  wedge  at  the  joint  of 
the  counter-brace,  a  permanent  strain  may  be  thrown  upon  the 
brace  equal  to  that  which  results  from  the  passage  of  the  maxi 
mum  load ;  this  strain  is  not  increased  by  the  passage  of  the 
load,  the  effect  of  which  is  simply  to  relieve  the  counter-brace, 
and  as  the  compression  of  the  brace  if  within  the  elastic  limits 
is  in  no  respect  injurious,  but  on  the  contrary  highly  conducive 
to  stiffness,  it  follows  that  the  compression  of  the  counter-brace 
to  an  extent  sufficient  to  throw  a  strain  upon  the  brace  equal 
to  that  which  results  from  the  passage  of  the  maximum  load, 

*  See  description  of  the  improved  lattice. 


WOODEN    BRIDGES.  87 

is  not  only  admissible,  but  very  desirable.  This  strain  is  the 
greatest  that  can  be  thrown  upon  a  counter-brace ;  the  passage 
of  a  load  relieves  instead  of  increasing  it,  and  it  will  be  safe 
to  calculate  the  size  by  the  condition  that  it  shall  produce  the 
required  compression  on  the  brace. 

Let  A  B  Q  D  be  a  rectangle,  and  suppose  a  force  to  act  in 
the  direction  of  the  diagonal  A  (7,  0  being  a  fixed  point. 

FIG.  54. 


If  the  intensity  of  the  force  be  represented  by  A  07  the 
components  in  the  direction  of  the  sides  will  be  A  D  and  A 
B,  and  those  which  result  from  the  resistance  of  the  fixed 
point  (7  will  be  QD  and  OB.  These  four  components  pro 
duce  a  force  of  extension  on  the  diagonal  D  B,  the  magnitude 
of  which  is  represented  by  D  B.  This  is  the  measure  of  the 
force  which  must  be  produced  by  wedging  the  counter-brace  ; 
and  as  this  diagonal  is  equal  to  A  (7,  it  follows,  that  the  strain 
upon  the  counter-brace  which  produces  a  given  pressure  upon 
the  brace,  is  equal  to  that  pressure  itself. 

If  w  represent  the  greatest  weight  that  can  ever  press  upon 
any  point  of  a  bridge,  W  X  Sec.  B  AC  will  be  a  little  greater 
than  the  strain,  and  in  practice  may  be  taken  to  represent  it. 

As  the  greatest  accidental  weight  that  can  ever  act  at  a 
single  point  is  very  small  when  compared  with  the  uniform 
load,  it  follows,  that  counter-braces  may  be  very  small  when 
compared  with  other  timbers. 

To  determine  the  strain  upon  the  braces  and  ties. 

To  estimate  the  strain  upon  the  brace  D  /,  we  may  suppose 
the  whole  of  the  bridge  between  A  and  D  to  be  suspended  at 
the  point  Dy  and  the  measure  of  the  force  would  be  that  which 


BRIDGE   CONSTRUCTION. 


would  hold  it  in  equilibrium ;  but  in  estimating  the  weight,  it 
is  not  sufficient  to  take  simply  the  weight  of  the  structure  it 
self,  to  this  must  be  added  the  greatest  load  that  could  ever 
come  upon  it. 


In  a  railroad  bridge,  the  greatest  load  is  probably  when  a 
train  of  loaded  cars  occupies  the  entire  space  between  A  and 
D,  and  the  driving  wheels  of  the  locomotive  are  directly  over  D. 
The  weight  of  the  cars  may  be  regarded  as  a  uniform  load 
distributed  over  A  D,  and  its  centre  of  gravity  would  be  at  a 
distance  of  one-half  A  D  from  the  point  of  rotation  E.  The 
weight  on  the  driving  wheels  of  the  locomotive  may  be  con 
sidered  as  acting  with  a  leverage  equal  to  the  whole  distance 
A  D.  Let  the  weight  of  the  bridge  between  A  and  D  be 
represented  by  TF",  the  uniform  load  on  A  D  by  w',  and  the 
weight  on  D  by  w"  ';  then  taking  the  sum  of  the  moments, 
we  have  w  (J-  A  D)  +  ivr  (J  A  D)  +  w"  (AD)=fAD  or 
iv  _j_  wi  _i_  2  w" 

—  9  —       —  =f  =  the  vertical  force  which  applied  at  D 

will  sustain  the  load. 

The  strain  upon  the  brace  will  be  very  nearly  f  sec.  mD 


When  there  are  intermediate  braces  and  ties,  as  p  p',  it  will 
not  vary  much  from  the  truth  to  suppose  the  strain  which  was 
thrown  upon  a  single  brace  in  former  case  by  the  uniform  load 
to  be  divided  equally  amongst  all  that  the  interval  contains. 
If  one  intermediate  tie  be  introduced,  it  will  bear  one-half,  if 

two,  each  one-third,  if  n,  -  -—  ^  part  of  the  uniform  load,  and 

~~ 


~\~ 


W  + 

this  is  expressed  by  o~?  — 


Tv 


WOODEN    BRIDGES.  89 

The  weight  on  the  driving  wheels  of  the  locomotive  being 
applied  at  a  single  point,  could  not  be  regarded  as  divided 
amongst  all  the  intermediate  braces  of  the  interval  3  D. 
When  this  weight  is  at  p,  the  brace  p  pf  will  sustain  more  of 
the  pressure  than  Sm,  or  D  f.  The  proportion  will  depend 
on  the  stiffness  of  the  chord  and  the  compressibility  of  the 
brace,  and  must  be  determined  upon  the  principles  by  which 
we  ascertain  the  pressure  of  a  beam  laid  over  several  supports. 

The  problem  for  determining  the  strain  thrown  upon  a  par 
ticular  brace  by  the  passage  of  a  variable  load,  is  very  com 
plicated  in  its  practical  application,  and  in  consequence  of  de 
fects  of  workmanship,  little  confidence  can  be  placed  in  the 
results. 

Even  if  we  suppose  the  whole  weight  w  to  be  thrown  upon 
the  intermediate  brace  without  the  adjacent  ones  sustaining 
any  portion  of  it,  the  area  will  not  be  increased  more  than  from 
three  to  five  square  inches  above  the  true  dimensions  ;  and  in 
practice  this  allowance  might  be  made  without  increasing  much 
the  size  of  the  timbers,  although  it  is  evidently  more  than 
sufficient. 

Of  the  strain  upon  the  ties  and  braces  at  the  centre. 

We  have  seen,  in  examining  the  forces  which  act  upon  a 
single  beam  supported  at  the  ends  and  uniformly  loaded,  that 
there  exist  both  horizontal  and  vertical  forces  at  every  point 
except  at  the  middle  and  at  the  extremities.  At  the  middle, 
when  the  load  is  uniform,  the  strains  are  altogether  horizontal, 
the  two  parts  being  exactly  balanced,  mutually  support  each 
other,  and  no  vertical  strain  is  experienced ;  but  at  other  points, 
the  distances  to  the  extremities  being  unequal,  the  pressure  of 
one  part  will  be  greater  than  that  of  the  other,  the  horizontal 
strains  will  no  longer  balance,  and  the  difference  must  be  com 
pensated  by  the  vertical  resistance  produced  by  a  cross  strain 
upon  the  fibres. 

This  vertical  force  increases  from  the  centre,  where  it  is 
zero,  to  the  extremities,  where  it  is  equal  to  one-half  the  whole 
uniform  weight  upon  the  bridge ;  and  the  increase,  when  the 
weight  is  uniform,  is  proportional  to  the  distance  from  the  centre. 


90  BRIDGE   CONSTRUCTION. 

In  a  bridge,  the  office  of  the  chords  is  to  resist  the  horizontal 
forces,  and  that  of  the  ties  and  braces  the  vertical  forces,  and 
as  the  strain  resulting  from  the  uniform  load  is  zero  at  the  cen 
tre,  it  follows,  that  the  sizes  of  the  intermediate  timbers  may  be 
much  smaller  here  than  at  the  abutments,  as  they  have  very 
little  more  strain  to  bear  than  that  which  results  from  the  por 
tion  of  the  variable  load,  which  acts  immediately  over  them, 
which,  in  a  long  span,  is  comparatively  trifling. 

Each  successive  brace,  in  passing  from  the  centre  to  the 
abutment,  is  more  and  more  strained,  and  consequently  should, 
if  properly  proportioned,  be  increased  in  size,  but  as  such  in 
crease  would  add  greatly  to  the  expense  and  trouble  of  framing, 
it  is  preferable  in  practice  to  make  all  the  timbers  uniform  and 
compensate  for  the  additional  strain  at  the  ends  by  additional 
braces  called  arch  braces. 

As  the  preceding  method  of  investigation  might  be  con 
sidered  objectionable,  and  doubts  be  entertained  of  the  correct 
ness  of  the  important  consequences  which  result  from  it,  on 
the  ground  that  the  analogy  is  not  perfect  between  a  beam 
supported^  at  the  ends  and  the  framed  truss  of  a  bridge,  we 
will  endeavor  to  present  a  different  view  of  the  subject. 

The  important  principle  that  we  aim  to  establish  is,  that  a 
great  difference  exists  between  the  strains  on  the  ties  and 
braces  at  the  centre  and  at  the  ends,  the  precise  law  of  increase 
or  diminution  is  of  secondary  importance,  and  will  not  now  be 
considered.* 

It  has  been  stated  that  when  a  truss  settles,  the  rectangles 
formed  by  the  ties  and  chords  become  oblique  parallelograms, 
the  diagonal  in  the  direction  of  the  brace  being  compressed  and 
the  opposite  one  extended.  Could  we  ascertain  the  exact  de 
gree  of  reduction  which  the  length  of  the  brace  experienced,  we 
would  have  a  certain  measure  of  the  strain.  To  determine 
this  by  calculation  would  be  difficult,  as  the  change  of  figure 

*  This  paragraph  was  penned  about  eight  years  ago,  at  which  time 
the  writer  was  not  aware  that  any  bridges  had  been  constructed  in  such 
a  way  as  to  recognize  the  existence  of  this  increased  strain  at  the  abut 
ment.  But  in  several  of  the  plans  that  are  now  in  use  the  principle 
leems  to  have  received  attention. 


WOODEN   BRIDGES.  91 

would  be  affected  by  the  form  of  the  curve  of  flexure,  and  the 
changes  in  the  lengths  of  the  sides  of  the  rectangles. 

FIG.  56. 


We  can  however  approximate  to  the  result  sufficiently  near 
to  confirm  the  previous  conclusions. 

Regarding  the  curve  of  flexure  when  very  slight  as  a  circle, 
draw  lines  to  all  the  angular  points,  and  A  T  a  tangent  at  the 
point  A.  Call  a  —  angle  T  A  b  and  n  =  the  whole  number  of 
intervals  A  6,  b  c,  c  d.  As  the  chords  are  all  equal,  the  angles 
at  A  would  also  be  equal,  and  the  angle  TA  B  =  n  a,  which 
measures  the  angle  of  change  of  the  first  rectangle,  is  n  times 
as  great  as  Tf  e  d  =  a,  which  measures  the  angle  of  change  of 
the  rectangles  in  the  centre.  Now,  for  small  deflections,  the 
diminution  in  length  will  be  proportional  to  the  angle,  and,  as 
this  diminution  measures  the  strain,  it  follows,  that  it  must  be 
n  times  as  great  at  the  abutments  as  at  the  centre.  "When  n 
becomes  infinite,  as  we  may  consider  it  in  a  beam,  a  =  o,  and 
the  strain  at  the  centre  is  nothing. 

The  changes  in  length  of  the  chords  in  the  central  interval 
do  not  affect  the  diagonal,  as  they  compensate  each  other. 

The  strain  at  the  abutment  is  constant  whatever  may  be 
the  length  of  the  intervals,  as  it  is  measured  by  the  tangent 
B  A  T:  results  which  correspond  with  those  obtained  by  the 
other  method. 

Effects  upon  the  braces  and  ties  which  result  from  the  intro 
duction  of  Arch-braces. 

In  a  system  of  this  kind  it  is  necessary  to  estimate  the  strains 
by  dividing  the  truss  into  parts,  and  considering  the  action  of 
each  part  separately. 


92 


BRIDGE   CONSTRUCTION. 


Let  d  D  and  e  E  represent  two  arch-braces  extending  from 
the  points  d  and  c  near  the  centre  to  the  abutments.  If  we 
disregard  compression,  the  effects  of  which  will  be  considered 
subsequently,  it  is  evident  that  the  weight  of  the  portion  dc 
may  be  regarded  as  suspended  from  the  points  d  and  c,  and 
will  be  entirely  transmitted  to  the  abutments,  exerting  no  in 
fluence  whatever  upon  the  parts  A  d  and  c  B. 

FIG.  57. 


\ 


If  a  D  bo  another  arch-brace,  the  portion  a  d  will  be  sus 
pended  from  a  and  d,  and  its  weight  transmitted  to  the  abut 
ments  by  a  D  and  d  D. 

The  calculation  of  the  strains  in  this  case,  therefore,  be 
comes  extremely  simple  ;  we  can  regard  the  whole  weight  of 
the  bridge  as  supported  by  the  arch-braces,  and  the  load  upon 
the  ordinary  braces  will  be  only  that  which  is  due  to  the  small 
intervals  ad,dc;  at  d  for  example  the  strain  upon  the  or 
dinary  brace  would  be  one-half  the  weight  on  d  £,  at  a  it  would 
be  one-half  the  weight  on  a  d,  and  it  therefore  follows,  that  by 
the  introduction  of  arch-braces  of  sufficient  size  to  make  their 
compression  inconsiderable,  the  ordinary  timbers  may  be  re 
duced  to  very  small  dimensions. 

The  weight  upon  any  arch-brace  (dD)  is  one-half  the  load 
upon  (a  d  +  d  c).  Call. this  weight  w,  and  let  D  E  =  s,  d  n  = 
h,  Dn  =  m,  n  E  =  mf,  and  I  =  length  of  brace  D  d; 


Then   8  :  m'  :  : 

transmitted  to  D. 

w 
Also  him::- 


wm 


w 


m1 


=  d  o  =  portion    of  the    weigh* 


mm 


:  dp  =  w  — j-  =  strain  upon  the  brace 
s  fii 

b  D.     This  strain  is  a  maximum  when  m  =  mr  or  when  b  is 
at  the  centre. 

We  cannot  however  regard  the  arch-brace  as  incompressi 
ble  ;  on  the  contrary,  it  is  known  that  timber  will  admit  of  a  re- 


WOODEN    BRIDGES.  93 

duction  of  .001  of  its  length  without  impairing  its  elasticity, 
and  therefore  an  arch-brace  100  feet  long  would  admit  of  a  re 
duction  of  li  inches  without  injury. 

The  effect  of  this  compression  would  be  to  throw  a  portion 
of  the  strain  upon  the  ordinary  ties  and  braces  depending  upon 
the  relative  stiffness  of  the  truss,  and  the  compressibility  of  the 
arch-brace. 

As  the  truss  is  relieved  in  proportion  to  the  amount  of  strain 
thrown  upon  the  arch-brace,  the  introduction  of  wedges  at  the 
extremity  would  furnish  the  means  of  regulating  this  propor 
tion  at  pleasure,  and,  if  necessary,  the  arch-braces  could  be 
made  to  sustain  the  whole  of  the  weight. 

It  may  not  be  improper  at  this  place  to  advert  to  an  objec 
tion  which  is  sometimes  made  to  a  combination  of  two  distinct 
systems  in  the  same  truss.  It  is  said  that  the  workmanship 
can  never  be  so  perfect  that  both  systems  will  assist  in  sus^ 
taining  the  load,  and  that  either  one  or  the  other  will  sustain 
the  whole.  This  remark  was  made  by  a  gentleman  who 
deservedly  ranks  amongst  the  first  in  his  profession  as  a  Civil 
Engineer,  and  who  is  the  inventor  of  a  plan  of  bridge  con 
struction  which  the  writer  regards  as  one  of  the  most  scientifi 
cally  proportioned  of  any  in  general  use.  That  it  is  erroneous 
we  think  can  be  readily  shown. 

Were  the  materials  of  a  bridge  perfectly  incompressible,  or 
even  nearly  so,  the  remark  would  be  correct,  as  one  system 
would  break  before  the  other  could  be  brought  into  action. 
But  wood  is  highly  elastic,  and  admits  of  a  considerable  exten 
sion  or  reduction  of  length  without  injury ;  consequently,  if  at 
the  instant  of  striking  the  false  works,  one  system  should  be 
overloaded,  it  would  settle  more  than  the  other,  and  the  second 
be  thus  brought  into  action.  The  strain  upon  the  twro  might 
not  be  precisely  equal;  but  this  is  of  little  practical  conse 
quence. 

Experience  confirms  this  view :  many  of  the  bridges  on  the 
public  works  of  Pennsylvania,  having  been  too  lightly  propor 
tioned,  settled  greatly ;  they  were  strengthened  by  the  addition 
of  arch-braces  or  arches,  and  have  since  stood  well.  If  then  the 
introduction  of  an  independent  system,  after  a  truss  has  com- 


94 


BRIDGE   CONSTRUCTION. 


menced  to  yield,  can  arrest  its  progress,  it  cannot  be  doubted 
that  the  effect  would  be  still  more  beneficial  if  introduced  at 
the  time  of  its  construction. 

Vo  determine  the  strains  upon  the  chords. 

The  maximum  strain  upon  the  chords  of  a  straight  bridge 
is  in  the  centre ;  being  one  of  compression  on  the  upper  chord, 
and  of  extension  on  the  lower,  its  magnitude  is  represented 
by  a  force  which,  applied  horizontally  at  A,  would  keep  the 
half  truss  A  B  from  falling. 

If  the  bridge  be  uniformly  loaded,  the  centre  of  gravity  will 
be  vertically  under  n,  the  middle  point  of  A  B,  and  if  w  repre 
sent  the  uniform  weight,  its  moment  in  reference  to  the  point 
of  rotation  0  will  be  w  X  m  c,  or  if  s  represent  the  span  it  will 

W  8 

be  —r .     An  accidental  load  will  produce  the  greatest  strain 

wr  8 

at  the  centre;  its  leverage  will  then  be  J  s  and  w'  X  -J  s  =  -^- 

=  moment  of  the  accidental  load. 

FIG.  58. 


The  sum  of  these  moments  will  be 


s. 


The  horizontal  force  at  A  acts  with  a  leverage  B  O  =  h,  its 

w  4-  2  wr 
moment  will  therefore  be  Hh.     Hence  ff=  — j-^ —  s  —  the 

force  wrhich  measures  the  tension  at  the  lower,  and  the  com 
pression  at  the  upper  chord. 

The  same  result  would  be  obtained  by  referring  the  mo 
ments  to  a  neutral  axis  passing  through  the  middle  of  the  truss ; 
and  although  not  quite  so  simple,  this  is  in  some  respects  the 
best  way  of  considering  the  question,  as  there  are  in  fact  two 
forces,  one  at  the  upper,  the  other  at  the  lower  chord,  acting 


WOODEN  BRIDGES.  95 

v;th  a  leverage  =  J  h.     The  analogy  of  the  truss  to  a  beam 
supported  at  the  ends  is  thus  preserved. 


FIG.  59. 

p          s         B 


If  we  suppose  a  single  force  acting  at  A  to  keep  the  truss 
in  equilibrium,  0  being  a  fixed  point  and  (6r)  the  centre  of 
gravity,  we  may  transfer  the  force  at  A  to  the  point  p  of  its 
line  of  direction,  and  the  two  forces  p  B  and  p  6r  will  have  a 
resultant,  which,  in  the  case  of  equilibrium,  will  pass  through 
the  point  (7,  and  p  0  will  represent  the  direction  of  the  force. 

"VVe  can,  however,  estimate  the  moments  in  a  different  man 
ner,  and  one  which,  although  it  will  give  the  same  result,  will 
express  better  the  true  conditions  of  the  problem. 

Instead  of  one  force  there  are  two,  one  at  A,  the  other  at  D. 
The  effect  of  these  forces  would  be  to  cause  rotation  around  the 
middle  of  the  line  joining  their  points  of  application.  The 
locus  of  the  point  of  rotation  must  therefore  be  in  the  line  or  n. 
But  the  weight  of  the  truss  reacts  upon  the  fixed  point  (7,  and 
generates  a  resistance  which  can  be  replaced  by  a  force  acting 
upwards.  There  are,  therefore,  two  vertical  as  well  as  two 
horizontal  forces,  one  acting  upwards  at  (7,  the  other  down 
wards  through  the  centre  of  gravity  Q-.  As  these  forces  are 
equal,  the  locus  will  be  in  the  line  s  s  which  bisects  6r  n,  and 
the  intersection  of  the  two  loci  s  sf  and  o'  n  will  give  0,  the  true 
point  of  rotation  of  the  four  forces.  The  resultant  in  the  case 
of  equilibrium  passes  through  o  and  (7,  and  evidently  coincides 
with  the  line  c  p  as  first  determined. 


96  BRIDGE    CONSTRUCTION. 

If  we  suppose  the  truss  to  be  without  a  horizontal  tie,  the 
conditions  of  equilibrium  are  in  no  manner  affected  :  the  re 
sistance  of  the  abutment  at  0  supplies  the  place  of  the  tension 
at  D,  and  the  direction  of  the  resultant  Op,  which  represents 
the  pressure  upon  the  abutment,  will  be  determined  as  before. 

Should  it  be  desirable  to  construct  a  curved  truss  on  the 
principles  of  the  arch  of  equilibrium,  instead  of  making  0  0  the 
proper  figure  for  this  curve,  it  would  be  better  to  bolt  it  to  the 
truss  in  the  direction  of  the  line  An  0.  00  could  then  be 
made  of  any  form  that  would  produce  the  best  architectural 
effect,  and  the  curve  of  equilibrium,  in  consequence  of  the 
greater  rise,  would  be  much  increased  in  strength.  The  curve 
of  equilibrium  is  useful  only  where  the  road  is  constant,  or  the 
variable  load  relatively  very  small. 

The  weight  of  a  structure  can  be  ascertained  from  the  bill 
of  materials ;  and  the  readiest  way  of  determining  the  position 
of  the  centre  of  gravity  in  any  intricate  combination  is  by 
means  of  a  model.  It  is  not  however  necessary  in  ordinary 
cases  to  resort  to  this.  The  weight  of  the  roadway,  as  also  the 
maximum  load  upon  it,  is  nearly  uniformly  distributed,  so  also 
are  the  weights  of  the  chords ;  the  only  increase  in  weight 
towards  the  abutments  is  due  to  the  increased  lengths  of  the 
ties  and  braces,  which  being  very  small  when  compared  with  the 
other  weights,  can  affect  but  slightly  the  position  of  the  centre 
of  gravity ;  and  as  the  small  error  is  in  favor  of  stability,  it  is 
in  almost  all  cases  proper  to  estimate  the  centre  of  gravity  at 
a  distance  from  the  abutment  equal  to  one-fourth  the  span. 

Where  the  chords  are  of  considerable  depths,  it  becomes 
necessary  to  take  into  consideration  the  distance  from  the 
neutral  axis. 

FIG.  61. 


As  the  strains  vary  as  the  distance  from  the  neutral  axis, 
if  we  represent  the  strain  at  0  by  On,  and  join  n  s,  onf  will 


WOODEN    BRIDGES.  97 

represent  the  strain  at  o  and  the  whole  resistance  of  the  chord 
will  be  represented  by  the  area  of  the  trapezoid  On  nf  o  mul 
tiplied  by  the  distance  of  the  centre  of  gravity  from  s. 

The  centre  of  gravity  will  always  fall  nearly  in  the  middle 
of  C  o.  Even  when  0  o  =  o  s,  the  error  will  be  only  T1a  of 
0  o,  by  considering  it  in  the  centre,  and  in  ordinary  cases  it 
will  not  be  •£•$.  The  error  is,  moreover,  in  favor  of  stability. 

"We  may  therefore,  in  practice,  consider  the  centre  of  gravity 
as  falling  in  the  middle  of  C  o,  and  the  leverage  will  be  the 
distance  from  this  point  to  s.  The  average  strain  upon  the 
joint  is  represented  by  a  5,  and  is  determined  from  the  propor 
tion  s  e  :  s  a  :  :  R  i  ab  =  R  — .  R  representing  the  maxi 
mum  strain  that  it  is  considered  safe  to  allow. 

Sometimes  two  or  more  chords  are  used  at  different  dis 
tances  from  the  neutral  axis. 


FIG.  62. 


o._ kl_ _ o 

Let  .A  5  and  Q  D  represent  two  chords,  at  distances  a  0 
and  c  0  from  the  neutral  axis  o  o'.  Draw  a  5,  to  represent  the 
maximum  strain  proper  to  allow  at  the  centre  of  the  upper 
chord,  and  join  b  c,  c  d  will  then  represent  the  strain  upon  the 
second  chord,  which  accordingly  opposes  much  less  resistance 
than  the  first. 

As  this  case  often  occurs,  particularly  in  the  construction 
of  ordinary  lattice  bridges,  it  may  be  satisfactory  to  give  the 
equation  of  moments.  Let  R  =  strain  per  square  inch,  at  the 
distance  a  <?,  a  =  area  in  inches  of  the  section  of  each  of  the 

Si 

four  chords,  d  =  a  c,  c  =  c  0.    Then  R  —  —  C  d—  strain  per 

ct 

square  inch  on  second  chord,     a  R  x  d  =  moment  of  first 
chord,     a  R  -y  X  c  =  moment  of  second  chord. 


98  BRIDGE   CONSTRUCTION. 

n  2  W  8 

a  R  (d  +  -7)  =  -7T  is  the  equation  of  moments,  from  which 

the  strain  upon  the  chords  with  a  given  weight  can  be  calcu 
lated,  or,  the  strain  being  assumed,  the  size  of  the  cross-section 
(a)  can  be  ascertained.  In  this  expression  w  —  whole  uniform 
weight,  s  =  span. 

If  a  truss  be  constructed  ivitli  parallel  arches,  the  strains 
upon  each  will  be  calculated  upon  the  same  principles. 

FIG.  63. 


Let  A  of  and  B  0  be  two  curved  chords  or  arches.  The 
resistance  of  the  abutment  at  (7,  takes  the  place  of  the  strain 
upon  the  lower  chord,  in  a  straight  bridge,  and  the  neutral 
axis  will  still  exist  half  way  between  A  and  S.  The  horizon 
tal  strains  varying  as  the  distance  from  the  neutral  axis,  it 
follows,  that  the  two  arches  can  never  sustain  equal  portions 
of  the  strain;  but  if  one  of  them,  as  in  the  figure,  intersects  the 
line  A  S  at  its  middle  point,  it  will  not  assist  in  the  slightest 
degree  in  sustaining  the  strain  at  that  part  of  the  truss,  which 
will  be  thrown  entirely  on  the  upper  arch  at  the  point  A. 

At  the  abutments  the  condition  of  things  is  reversed ;  the 
lower  chord  sustains  a  horizontal  strain  equal  to  that  at  the 
centre,  and,  in  addition,  a  vertical  force  resulting  from  the 
weight  of  the  half  truss.  The  resultant  of  the  two  is  an  ob 
lique  line  c  n,  determined  by  joining  0  with  the  point  of  inter 
section  of  a  horizontal  line  through  A,  and  a  vertical  through 
the  centre  of  gravity  (7.  The  end  o'  of  the  upper  chord  sus 
tains  no  pressure. 

This  case  is  one  in  which  apparent  strength  is  real  weak 
ness  ;  one  of  the  arches  at  the  centre,  and  the  other  at  the  abut 
ments,  contribute  nothing  towards  sustaining  the  horizontal 
strains.  A  single  arch  extending  from  A  to  <7,  would  give 


WOODEN   BRIDGES.  99 

the  same  strength  with  half  the  material,  when  the  dimensions 
are  uniform  throughout. 

That  a  system  similar  to  that  represented  in  the  figure 
should  be  properly  proportioned,  the  upper  arch  should  dimin 
ish  from  the  centre  to  the  end,  and  the  lower  one  from  the  end 
to  the  centre. 

We  will  conclude  this  part  of  the  subject  with  a  practical 
exemplification  of  the  manner  in  which  the  sizes  of  the  chords 
are  calculated. 

As  our  object  is  merely  to  illustrate  a  principle,  great  accu 
racy  will  not  be  attempted,  and  round  numbers  only  will  be 
used. 

FIG.  64. 


Let  x  =  area  of  the  cross  section  of  the  arch  in  inches 
s  =  span  =  500  feet 
w  =  whole  weight 

s 
-J-Q  =  rise  of  arch  —  50  feet 

H  =  1000  Ibs.  =  maximum  strain  per  square  inch. 

,  s        w       s  10  10 

^*        =    -  x 


The  greatest  variable  load  is  generally  considered  as 
caused  by  a  crowd  of  people,  and  is  estimated  at  120  Ibs.  per 
square  foot. 

If  the  bridge  be  supposed  20  feet  wide,  and  supported  by 
four  trusses,  each  will  bear  5  X  120  =  600  Ibs.  per  foot  linear. 
The  weight  of  the  structure  must  be  determined  by  assum 
ing  the  dimensions  of  all  its  parts,  making  out  a  bill  of  mate 
rial,  and  finding  its  weight  from  a  knowledge  of  its  cubic  con 
tent  and  specific  gravity.  At  present,  we  will  assume  the 
weight  of  the  structure  to  be  equal  to  the  greatest  lead,  600 
Ibs.  per  foot. 

We  then  have  W=  1200  x  500  =  600,000  Ibs. 


100  BRIDGE   CONSTRUCTION. 

The  horizontal  strain  at  0  is  R  x  =  ~-  =  750,000  and  a 

o 

=  750  square  inches,  or  about  27  inches  square  for  the  size  of 
the  arch  at  the  centre. 

/H       ? 
w       An  /100  +  16 

We  have  also  77  X  =  strain  at  A  =  \  / x 

2        no  v        s 

10 

|  =  |  VH6  =  802,500  Ibs.  or  802  square  inches,  about  29 J 

inches  square  for  the  size  of  the  arch  at  the  abutments. 

When  two  independent  systems  are  combined  in  the  con 
struction  of  a  truss,  it  becomes  difficult,  if  not  practically  im 
possible,  to  estimate  the  portion  of  the  strains  sustained  by 
each,  owing  to  the  defects  in  mechanical  execution  insepara 
bly  connected  with  every  structure.  If  a  calculation  be 
attempted,  it  can  only  be  upon  the  supposition  that  the  joints 
are  absolutely  perfect,  and  that  at  the  first  instant  of  flexure 
both  systems  are  in  full  bearing,  and  oppose  a  resistance  pro 
portionate  to  their  relative  stiffness. 

Disregarding  the  particular  arrangement  of  ties  and  braces, 
or  the  greater  or  smaller  number  of  the  intervals,  we  will  con 
sider  the  trusses  as  acting  as  a  whole.  This  can  be  done  with 
propriety  on  the  supposition  that  the  joints  are  perfect,  and  a 
general  solution  of  the  problem  becomes  very  simple. 

If  the  trusses  act  as  a  whole,  the  deflections  may  be  con 
sidered  as  proportional  to  the  weights ;  but  the  strains  upon 
the  chords  are  as  the  weights  directly,  and  as  the  areas  of  the 
cross-sections  inversely,  and  the  deflections  must  therefore  be 
in  the  same  proportion. 

Let  a  represent  the  area  of  the  cross-section  of  the  chords 
of  one  system,  and  n  a  that  of  the  other :  the  depth  and  length 
of  truss  in  each  being  equal.  If  d  represent  the  deflection  of 
the  first  system,  with  a  given  weight,  n  d  will  express  the  de 
flection  of  the  second. 

Let  x  represent  the  actual  deflection,  which  is  of  course 

w  x 
equal   in   both.      Then   d  :  x  :  :  w  :  — -5—  =  weight   on   first 


WOODEN   BRIDGES.  101 

fl/i    />• 

system,  and  n  d  :  x  : :  w  :  — -j-  =  weight  on  second   system. 

The  sum  of  these  must  equal  the  whole  weight.     Hence 

w  x       w  x  nd          ,  w  x 

w  —  — -r  +  —7—.     Whence  x  —  —  —=-  and  — 7—  = 
nd         d  n  4-  1  d 

.     n  iv  x  1 

(—  7- =-j  w  =  weight  on  first  system,  — r  = =•  t0 

vra  +  1'  >  w  ^        n  +  1 

=  weight  on  second  system. 

In  other  words,  the  strain  upon  each  system  will  be  ex 
actly  proportioned  to  its  powers  of  resistance,  and  the  whole 
together  may  be  estimated  as  one  truss. 

In  the  construction  of  a  bridge  with  a  system  of  arch- 
braces,  the  simplest  and  best  plan  is  to  depend  upon  the  latter 
to  sustain  the  entire  weight  of  the  structure,  using  only  a  very 
light  truss  with  counter-braces  or  diagonal  ties  to  establish  the 
necessary  connection  of  the  parts,  prevent  flexure  and  vibra 
tion,  and  resist  the  action  of  variable  loads. 

Instead  of  using  arch  braces,  trusses  are  sometimes  strength 
ened  by  the  addition  of  arches.  Great  benefit  results  from  their 
use,  but  nearly  the  same  effects  may  be  obtained  by  arch- 
braces. 

An  arch,  when  of  the  proper  figure  of  the  curve  of  equilib 
rium,  is  capable  of  sustaining  any  constant  load  without  change 
of  form  ;  but,  as  the  load  upon  a  bridge  is  variable,  it  is  ob 
viously  impossible  to  make  an  arch  of  equilibrium  for  a  wooden 
viaduct. 

The  flexibility  of  an  arch  renders  it  but  poorly  adapted  to 
sustain  a  variable  load ;  when  used  for  this  purpose,  therefore, 
it  must  always  be  connected  with  a  truss  capable  of  giving  it 
the  necessary  stiffness.  Such  combinations  are  extensively 
used. 

Means  of  increasing  the  strength  of  bridge  trusses. 

When  a  truss,  in  consequence  of  having  been  too  lightly 
proportioned,  gives  way  by  vertical  flexure,  an  arch,  or  arch- 
braces  with  a  straining-beam  connecting  the  upper  ends,  may 
be  bolted  to  the  truss.  Such  additions  have  been  often  made, 
and  are  found  to  answer  well.  Many  of  the  bridges  on  the 


102  BRIDGE    CONSTRUCTION. 

public  works  of  Pennsylvania  have  been  strengthened  in  this 
way,  and  rendered  sufficiently  strong  for  the  heaviest  locomo 
tives  and  trains. 

When  a  beam  is  laid  over  several  supports,  its  strength  for 
a  given  interval  is  much  greater  than  when  simply  supported 
at  the  ends.  The  same  principle  is  applicable  to  bridges,  and 
when  several  spans  occur  in  succession,  it  is  of  great  advantage 
to  continue  the  upper  and  lower  chords,  if  the  bridge  is  straight, 
across  the  piers.  By  this  arrangement,  the  strength  of  chords 
of  each  central  span  in  a  series  would  be  double  that  of  the 
same  spans  disconnected,  and  the  extreme  spans  would  be 
stronger  in  the  proportion  of  3  to  2. 

Notwithstanding  this,  we  often  see  bridges  in  which  the 
upper  chords  are  not  connected  over  the  piers,  and  the  absurd 
remark  has  been  made,  by  practical  builders,  that  the  bridge 
must  yield  somewhere,  and  better  there  than  elsewhere.  Just 
in  proportion  as  this  point  is  capable  of  opposing  a  resistance, 
must  the  strength  of  the  bridge  be  increased  ;  and  it  is  obvious 
that  if  a  bridge  should  be  cut  in  two  in  the  centre  of  the  span, 
and  one-half  removed,  the  other  half  could  not  fall  as  long  as 
the  connection  over  the  pier  remained  perfect. 

Even  in  bridges  of  a  single  span,  it  would  not  be  impossi 
ble  to  communicate  the  strength  of  a  continuous  bridge,  by 
connecting  the  upper  chords  with  chains  passing  over  the  back 
of  the  abutments,  and  anchored  into  the  ground  on  the  princi 
ple  of  a  suspension  bridge ;  but  such  an  arrangement  is  not  to 
be  recommended  in  ordinary  cases. 

When  the  chords  of  a  straight  bridge  are  of  equal  size,  the 
lower  are  necessarily  much  weaker  than  the  upper  within  the 
elastic  limits ;  the  latter  resist  a  force  of  compression  which 
naturally  closes  the  joints,  and  brings  every  part  of  the  cross- 
section  into  full  bearing.  But  the  case  is  very  different  at  the 
lower  chord  ;  here,  from  the  nature  of  the  strain,  which  is  one 
of  extension,  the  joints  are  opened,  and,  from  the  manner  in 
which  the  connection  is  formed,  only  one-half  the  area  of  the 
cross-section  opposes  any  resistance. 

Fortunately,  we  have  a  simple  means  of  correcting  the  evil ; 
but,  simple  as  it  is,  it  does  not  appear  to  be  generally  resorted 


WOODEN"    BRIDGES.  103 

to.  It  consists  in  placing  the  ends  of  the  lower  chords  in  close 
contact  with  the  abutment,  or,  which  is  still  better,  driving 
wedges  between  the  abutment  and  chords. 


FIG.  65. 


In  fact,  it  is  evident  that  if  the  truss  rests  against  an  abut 
ment  capable  of  opposing  at  horizontal  resistance,  the  tie  A  B 
could  be  cut  entirely  through  without  danger  of  the  truss  fall 
ing,  for  the  strain  upon  the  tie  at  0  is  exactly  replaced  by  the 
two  resistances  to  compression  at  A  and  B  ;  and  just  in  pro 
portion  as  a  pressure  can  be  produced  at  A  and  B,  in  exactly 
the  same  proportion  will  the  truss  be  relieved;  and  if  this  pres 
sure  should,  by  wedging  or  other  means,  be  made  to  exceed 
the  horizontal  thrust  of  the  truss,  the  centre  D  would  be  forced 
upwards. 

If  the  chord  A  B  should  be  slightly  curved,  as  represented 
by  the  dotted  line  A  Of  B,  the  result  would  be  the  same ;  the 
pressure  upon  the  abutment  would  not  become  almost  infinite, 
as  was  asserted  by  a  gentleman  with  whom  the  author  once 
corresponded  upon  the  subject  of  bridge  construction.  What 
ever  might  be  the  rise  0  6r/,  the  strain  at  A  and  B  would  not 
be  in  the  slightest  degree  affected  so  long  as  the  weight,  span, 
and  height  CD,  or  its  equal  n  B,  remained  constant. 

A  moderate  pressure  upon  the  face  of  the  abutment,  so  far 
from  being  an  injury,  is  a  very  decided  advantage,  as  it  serves 
to  counteract  the  pressure  of  the  embankment  on  the  other 
side,  and  allows  a  reduction  in  the  thickness  of  the  wall. 

For  the  reasons  assigned,  we  think  that  wedges  behind  the 
ends  of  the  lower  chords  of  straight  bridges  should  never  b« 
omitted. 


104  BRIDGE    CONSTRUCTION. 

On  tlie  maximum  span  of  a  wooden  bridge. 

It  requires  no  demonstration  to  prove  that,  in  order  that 
the  maximum  span  may  be  attained  in  a  bridge,  it  is  neces 
sary  that  every  part  should  be  properly  proportioned  to  the 
strain  that  it  may  be  required  to  bear.  The  strength  of  a  sys 
tem  is  the  strength  of  its  weakest  point ;  this  is  the  point  of 
fracture ;  and  any  increase  of  strength  at  other  points,  produced 
by  increasing  the  amount  of  material  beyond  its  minimum 
quantity,  only  increases  the  weakness  by  increasing  the  weight. 
It  follows,  therefore,  that  no  plan  which  does  not  distinctly 
recognise  this  principle  of  accurately  proportioning  the  dimen 
sions  to  the  strains,  and  apply  it  in  detail,  can  be  employed  for 
the  maximum  span. 

Many  large  bridges  have  been  constructed,  several  of  which 
have  considerably  exceeded  300  feet  in  span ;  but  in  all  these 
were  some  defects,  some  points  too  heavily  loaded  by  timbers 
of  unnecessarily  large  dimensions. 

Tredgold,  in  his  treatise  on  carpentry,  gives  a  plan  for  a 
bridge  of  400  feet  span,  the  support  of  which  consists  of  framed 
voussoirs,  as  they  are  termed ;  and  as  no  mention  is  made  of 
any  variation  in  size,  it  was  no  doubt  the  design  of  the  archi 
tect  to  make  the  dimensions  uniform. 


The  defects  of  this  arrangement  naturally  appear  from  the 
preceding  explanation  of  the  manner  in  which  the  pressures 
are  distributed,  varying  as  the  distance  from  the  neutral  axis. 

The  points  A  and  0  sustain  less  than  B  and  jD,  and  if  the 
sections  are  everywhere  the  same,  it  follows,  that  if  B  and  D 
are  sufficiently  strong,  A  and  0  must  possess  surplus  strength, 
and  with  it  unnecessary  weight  of  material. 


TTOODEN   BRIDGES.  105 

In  the  second  place,  the  diagonal  timbers  in  the  intervals 
add  considerably  to  the  weight,  without  corresponding  advan 
tage  ;  an  arch  is  very  well  able  to  resist  a  variable  load  upon 
one  side,  if  the  other  side  can  be  kept  from  rising ;  and  instead 
of  using  a  braced  arch,  it  is  better  to  make  the  truss  with  which 
it  is  connected  serve  as  a  counter-brace. 

A  system  which  appears  to  be  absolutely  the  lightest  and 
most  simple  that  could  be  used,  and  the  one  of  all  others  best 
calculated  to  attain  the  maximum  limit  of  the  span,  is  repre 
sented  in  the  following  figure. 

i 
FIG.  67. 


The  sole  support  of  this  truss  is  the  single  arch  A  B,  which 
increases  in  size  from  the  centre  to  the  ends  in  exact  propor 
tion  to  the  strain ;  n  nf  represents  the  line  of  roadway,  sup 
ported  by  uprights  from  the  arch,  and  between  these  uprights 
keyed  counter-braces  or  diagonal  bolts  are  introduced.  By 
wedging  the  counter-braces  or  screwing  the  nuts,  the  arch  can 
be  compressed  sufficiently  to  produce  a  strain  equal  to  that 
caused  by  the  maximum  load,  so  that  the  subsequent  passage 
of  a  load  will  only  relieve  the  counter-brace  without  adding  to 
the  weight  upon  the  arch,  and  the  upward  motion  at  the 
haunches  is  effectually  prevented  by  the  counter-bracing. 

The  hand  rail  on  the  top  of  the  bridge  may,  by  a  proper 
connection  with  the  truss,  be  made  to  assist  in  counter-bracing 
the  centre,  and  thus  every  part  performs  important  functions 
without  a  single  stick  being  superfluous. 

An  assertion  incidentally  made  in  the  above  description, 
perhaps  requires  further  elucidation.  We  have  said  that,  by 
wedging  the  counter-braces,  a  strain  can  be  thrown  upon  the 
bridge  equal  to  that  produced  by  the  maximum  variable  load, 
and  that  the  subsequent  passage  of  this  load  will  throw  nc 
additional  strain  or  weight  upon  the  arch. 


106  BRIDGE   CONSTRUCTION. 

The  importance  of  this  fact  renders  it  worthy  of  a  separate 
consideration. 

Effects  of  counter-bracing  upon  an  arch. 
FIG.  68. 


Let  As B  represent  an  arch,  supported  by  resisting  abut 
ments  at  the  points  A  and  B,  and  suppose  a  heavy  uniform 
load  to  be  distributed  along  the  roadway  CD.  The  effect  of 
this  load  is  to  depress  the  arch,  and  the  amount  of  the  depres 
sion  will  be  greatest  in  the  centre,  diminishing  towards  the 
abutments,  where  it  is  zero.  In  consequence  of  this  difference 
of  depression,  any  point  (n)  near  the  centre  will  sink  more 
than  (nf)  nearer  the  abutment ;  by  this  change  of  figure  the 
diagonal  n'  s  is  lengthened,  and  n  o  is  shortened.  Consequently, 
if,  before  the  application  of  the  weight,  the  counter-brace  n' 
s  was  exactly  in  contact  with  the  joint,  it  must  now  be  at 
such  a  distance  as  to  leave  an  interval. 

When  the  weight  is  removed,  the  arch  will  return  to  its 
original  figure,  and  the  interval  will  be  closed ;  but  if  wedges 
be  driven  into  all  the  intervals,  their  reaction  when  pressed 
will  prevent  the  arch  from  regaining  its  figure,  and  it  is  forci 
bly  held  in  the  position  in  which  the  load  placed  it,  and,  as  a 
necessary  consequence,  continues  subject  to  the  same  strain  as 
when  the  weight  was  upon  it.  It  certainly  needs  no  lengthy 
explanation  to  convince  any  one  that,  if  a  spring  or  a  beam  be 
bent,  either  by  a  weight  or  by  the  resistance  of  fixed  points,  if 
the  flexure  is  constant  the  strains  must  be  precisely  the  same, 
and,  consequently,  the  counter-braced  arch  is  not  more  strained 
when  the  weight  is  upon  it,  than  it  is  when  that  weight  is 
removed,  the  effect  of  the  weight  being  simply  to  relieve  the 
counter-braces,  which  are  not  strained  when  it  is  acting. 

If  any  should  regard  the  above  explanation  as  unsatisfac- 


TTOODEN   BRIDGES.  107 

tory,  and  be  unable  to  reconcile  the  apparently  paradoxical 
result,  that  a  heavy  weight  brought  upon  an  arch  produces  no 
strain,  the  following  considerations  niay  serve  to  remove  the 
difficulty. 

We  admit  that  the  proposition  is  not  strictly  true,  but  it  is 
nearly  so,  and  would  be  entirely  so,  if  the  counter-braces  and 
the  longitudinal  timber  with  which  they  are  connected  at  top 
were  incompressible ;  as  they  are  not,  the  arch  will  rise  a  little 
after  the  removal  of  the  weight,  but,  its  elastic  movements 
being  confined  within  very  narrow  limits,  great  stiffness  will 
be  secured. 

The  upward  action  of  the  arch,  in  consequence  of  its  elas 
ticity,  produces  a  force  which  presses  against  the  counter-braces, 
*nd  is  by  them  transmitted  to  the  longitudinal  timber  0  D, 
upon  which  it  produces  a  force  of  extension  exactly  the  reverse 
effect  to  that  which  is  produced  by  a  weight ;  and  when  the 
weight  acts,  this  strain  is  counteracted,  and  becomes  nothing. 

If  it  be  asked,  why  does  not  the  strain  upon  the  arch  throw 
an  additional  strain  upon  the  abutments?  The  answer  is,  that 
this  would  be  the  case  if  the  counter-braces  could  act  against 
fixed  points  not  connected  with  the  frame  itself;  but,  as  this  is 
not  and  cannot  be  the  case,  it  follows,  that  the  downward 
pressure  upon  the  arch  is  exactly  counterbalanced  by  an  up 
ward  pressure  at  the  other  end  of  the  counter-brace.  When  the 
weight  is  upon  the  bridge,  the  upward  pressure  is  counteracted, 
whilst  the  downward  pressure  is  as  before,  and  therefore  the 
pressure  upon  the  abutment  is  increased  by  exactly  this 
amount;  that  is,  by  the  accidental  weight,  whatever  it  may  be. 

Instead  of  counter-braces  of  wood,  diagonal  ties  of  iron, 
with  nuts  and  screws,  could  be  placed  in  the  direction  of  the 
other  diagonals ;  and  their  use  would  be  in  some  respects  very 
advantageous.  They  would  be  lighter,  and  consequently 
would  add  less  to  the  weight  of  the  structure.  The  strain 
upon  them  being  tensile,  they  could  be  of  any  desired  length 
without  danger  of  flexure,  to  which  counter-braces  are  liable ; 
and  the  degree  of  tension  could  be  very  conveniently  regulated 
by  means  of  the  screws,  without  the  danger  of  loosening,  which 
Is  connected  with  the  use  of  wedges. 


108 


BRIDGE   CONSTRUCTION. 


FIG.  69. 


A  truss  of  this  kind  with  a  parabolic  arch,  the  section  at 
every  point  being  proportional  to  the  strain,  and  protected  from 
the  effects  of  partial  loads  by  the  iron  diagonal  ties  m  n  wir  nf, 
&c.,  is  absolutely  the  lightest  that  we  can  conceive  for  a  wooden 
bridge,  fulfils  every  condition  of  a  perfect  structure,  and  con 
sequently  admits  of  the  greatest  possible  extension  of  the  span. 
If  a  horizontal  tie  is  desired,  the  posts  must  be  extended. 

Roadway. 

The  roadway  of  a  bridge  admits  of  little  variation.  It  is 
generally  constructed  by  laying  beams  across  the  trusses,  upon 
which  are  placed  the  longitudinal  pieces  which  carry  the  plank 
ing.  A  very  important  part  of  the  roadway  consists  in  the 
bracing,  which  is  necessary  to  prevent  lateral  flexure.  The 
usual  arrangement  of  braces  is  shown  in  the  annexed  figure. 

FIG.  70. 


A  B  and  0  D  are  the  trusses,  n  n'  the  girders,  and  the 
diagonal  timbers  are  braces. 
The  following  figure  represents  another  plan  of  horizontal 

FIG.  71. 


WOODEN   BRIDGES.  109 

bracing,  which  is  perhaps  the  lightest  that  could  be  used,  and 
would  be  well  adapted  to  large  spans,  where  the  quantity  of 
material  in  the  centre  is  required  to  be  the  least  possible.  The 
diagonal  timbers  between  the  arches  are  designed  to  be  em 
ployed  as  keyed  counter-braces. 

A  very  good  bracing  may  sometimes  be  obtained  by  spik 
ing  the  floor  plank  in  two  layers,  extending  diagonally  across 
the  bridge  and  crossing  each  other  at  right  angles. 


CAST-IRON  BRIDGES. 


IT  is  foreign  to  the  original  design  of  this  treatise  to  introduce 
the  subject  of  cast-iron  structures ;  but  as  the  same  general 
principles  must  guide  the  engineer  in  these,  as  in  other  bridges, 
a  paragraph  upon  the  subject  may  not  be  considered  out  of 
place. 

The  abundance  of  wood,  and  its  great  relative  economy, 
have  secured  its  adoption  in  this  country,  in  preference  to  iron  ; 
but  in  Great  Britain,  many  splendid  structures  have  been 
erected  of  the  latter  material,  which  possess  great  beauty, 
strength,  and  durability. 

If  the  principle  of  proportioning  every  part  to  the  strain 
which  it  has  to  bear,  is  important  in  its  application  to  timber- 
bridges,  much  more  must  it  be  when  applied  to  bridges  of  cast- 
iron  ;  for  the  expense  is  nearly  in  proportion  to  the  quantity  of 
material,  and  the  weight,  and  consequently  the  weakness,  is 
increased  by  every  pound  unnecessarily  added.  As  we  have 
said  already,  the  strength  of  a  bridge  is  the  strength  of  its 
weakest  point ;  and  of  course  the  accumulation  of  material 
where  it  is  not  needed,  so  far  from  being  of  advantage,  is  a 
positive  injury. 

It  is  therefore  of  the  first  importance,  in  designing  a  plan 
for  a  cast-iron  bridge,  to  place  the  material  which  is  to  resist 
the  horizontal  strains  at  the  greatest  possible  distance  from  the 

(110) 


CAST-IRON    BRIDGES.  Ill 

neutral  axis,  as  it  will  there  act  with  the  greatest  effect.  This 
object  is  secured  by  using  only  a  single  arch,  and  giving  it  the 
maximum  rise  that  the  nature  of  the  structure  will  admit. 
The  only  objection  that  can  possibly  be  made  to  a  single  arch, 
we  conceive  to  be  its  flexibility ;  but  if  it  can  be  so  counter- 
braced  as  to  prevent  a  change  of  figure  by  the  action  of  a  vari 
able  load,  we  cannot  perceive  that  any  thing  more  is  necessary. 
If  this  principle  be  correct,  it  follows,  that  most  of  the  plans 
which  have  been  used  are  to  some  extent  objectionable,  as 
they  consist  either  of  framed  voussoirs,  that  is,  of  two  parallel 
arches  separated  by  cross-braces,  or  of  several  arches  rising  at 
different  heights,  and  extending  to  different  elevations.  The 
latter  arrangement  would  perhaps  be  a  good  one,  where  the 
object  is  to  distribute  the  pressure  upon  many  points;  but  as  an 
abutment  can  always  be  made  sufficiently  strong  to  resist  the 
thrust  of  a  stone  arch,  it  cannot  be  supposed  that  there  would 
be  any  difficulty  in  guarding  against  the  pressure  of  a  much 
lighter  structure  of  cast-iron.  Certain  it  is,  that  all  the  arches 
cannot  act  at  the  same  distance  from  the  neutral  axis,  and 
therefore  a  smaller  quantity  of  material  at  the  maximum  dis 
tance  would  be  equally  efficient.  No  new  principle  is  in 
volved  in  the  construction  of  iron  bridges  ;  the  strains  are  dis 
posed,  and  must  be  guarded  against,  in  the  same  way  as  in 
wooden  structures ;  the  only  modifications  are  those  required 
by  the  peculiar  character  of  the  material,  and  by  the  greater 
difficulty  of  securing  proper  connections. 

FIG.  72. 


A  B  represents  an  arch  of  iron,  constructed  of  plates  of  con 
siderable  lengths,  laid  upon  each  other  so  as  to  break  joint,  and 
bolted  together,  or  in  any  other  suitable  way,  Co  being  the 
greatest  rise  that  can  be  given  to  the  arch. 

(a  «')  are  vertical  posts  or  columns,  which  may  be  either  of 
cast-iron  or  wood :  the  latter  woul  1  better  resist  an  impulsive 


112  BRIDGE   CONSTRUCTION. 

force,  owing  to  its  superior  elasticity ;  but  the  former  would  be 
more  elegant,  and  by  using  a  deep  string-piece  of  wood  (m  mf] 
as  a  cap,  there  would  probably  be  no  danger  of  fracture  from 
the  impulsive  force  of  a  passing  locomotive.  In  fact,  whatever 
may  be  the  plan  of  the  structure,  it  would  not  be  proper  to  place 
the  rails  of  a  railway  immediately  upon  cast  iron  supports ; 
there  should  be  some  elastic  substance  interposed  to  break  the 
force  of  shocks. 

It  is  probable,  therefore,  that  no  objection  could  be  made  to 
the  use  of  hollow  columns  for  the  posts  s  s'. 

A  s,  r  sf  are  rods  of  wrought-iron,  with  nuts  and  screws, 
designed  to  counter-brace  the  arch,  on  principles  already  ex 
plained,  and  prevent  it  from  rising  by  the  action  of  a  heavy 
variable  load  upon  the  opposite  side  or  at  the  centre. 

If  the  depth  of  the  arch  is  not  sufficient  to  prevent  the  cen 
tre  from  rising,  by  the  action  of  loads  upon  the  haunches,  the 
roadway  (m  mf)  may  be  raised  so  as  to  admit  of  diagonal  ties 
between  it  and  the  arch  at  0 ;  or,  in  some  cases,  the  handrail 
n  nr  may  be  so  connected  with  the  truss  as  to  form  a  very 
efficient  counter-brace,  or  a  slight  inverted  arch  could  connect 
the  interval  rr,  which  might  be  placed  below,  or  if  placed 
above  r  r,  it  might  form  part  of  the  railing  of  the  roadway ;  or, 
lastly,  a  straight  or  arched  piece  could  connect  pf  p9  and  the 
system  of  diagonal  ties  be  continued  to  the  centre. 

Such  a  combination  is  perhaps  the  lightest  that  could  pos 
sibly  be  made  to  span  a  given  interval. 

We  will  now  examine  the  effect  of  expansion. 

The  effect  of  expansion  will  evidently  be  to  cause  the  arch 
to  rise,  or  to  increase  the  versed  sine  o  0.  This  rise  will  dimin 
ish  towards  the  abutments,  where  it  becomes  nothing ;  but  the 
greatest  strain  upon  any  of  the  connecting  rods  will  be  at  the 
first  quadrilateral  A  s,  because  here  the  angular  change  of 
figure  will  be  greatest.  The  effect  of  the  change  is  to  strain 
the  tie  A  s,  but  to  compensate  for  this  is  the  expansion  of  the 
tie  itself,  by  the  same  change  of  temperature  which  affects  the 
arch  and  the  elasticity  of  the  beam  m  m',  which  will  stretch 
and  yield  to  the  strain  caused  by  the  extension  of  the  tie. 

The  greatest  extension  of  plate  or  bar  iron,  when  exposed 
to  the  extreme  variations  of  atmospheric  temperature,  is  '001 


CAST-IRON    BRIDGES.  113 

of  its  length  according  to  experiments  made  by  the  writer, 
and  the  extensibility  of  wood  according  to  Tredgold,  is  J= 


of  its  length,  without  injury.  From  these  data,  the  relative 
extensions  in  any  given  case  can  be  calculated.  In  an  arch 
of  500  feet  span,  and  50  feet  rise,  the  extension  amounts  to  -,% 
of  a  foot  or  6  inches. 

The  eifect  of  this  expansion,  if  the  truss  is  of  the  form  re 
presented  in  the  diagram,  is  more  than  half  counteracted  by 
the  expansion  of  the  ties  in  the  most  unfavorable  case,  and 
when  the  posts  which  support  the  roadway  are  not  very  far 
apart,  the  expansion  of  the  ties  may,  of  itself,  be  more  than 
sufficient  to  counteract  the  expansion  of  the  arch  ;  but  even  if 
it  should  not  be,  the  only  effect  would  be  to  extend  and  com 
press  laterally  the  wooden  beam  m  mf,  which  is  able  to  bear 
without  injury  four  times  the  extension  which  change  of  tern* 
perature  would  produce  upon  the  arch.  It  is  reasonable  to 
suppose,  then,  that  a  system  connected  in  this  way  would  have 
nothing  to  fear  from  changes  of  temperature. 

Other  forms  of  trusses  are  more  liable  to  be  affected  by 
changes  of  temperature,  and  it  is  important  in  arranging  the  de 
tails  of  an  iron  truss,  to  take  this  fact  into  consideration  ;  the 
extension  of  bar  iron  within  the  elastic  limits,  is  as  great  as  that 
caused  by  atmospheric  changes,  and  this  elasticity  is  in  general 
sufficient  to  effect  a  compensation,  and  prevent  any  injury  from 
excessive  strains.  The  principle  of  the  counter-braced  arch 
seems  to  be  peculiarly  well  adapted  to  the  construction  of  iron 
bridges,  as  the  compensation  is  almost  perfect,  and  the  only 
effect  of  expansion  or  contraction  will  be,  to  raise  or  depress 
very  slightly  the  crown  of  the  arch. 

Arches  composed  entirely  of  cast-iron  have  been  much 
used  for  bridges  in  England,  but  the  author  does  not  place 
much  confidence  in  tbe  material,  where  it  is  liable  to  be  sub 
jected  to  impulsive  forces  ;  an  arrangement,  which  he  con 
siders  far  preferable,  and  which  has  been  adopted  for  two  of 
the  bridges  on  the  Pennsylvania  Railroad,  consists  of  rolled 
plates  laid  one  upon  another  so  as  to  break  joint,  and  clamped1 
together,  with  or  without  a  centre  rib  of  cast-iron. 
8 


APPLICATION  OF  RESULTS. 


THE  results  deducible  from  the  preceding  general  theory  of 
Bridges,  will  now  be  condensed  and  practically  applied,  to  de 
termine  the  proportions  of  the  parts  of  a  bridge  of  assumed 
dimensions. 

In  proportioning  the  parts  of  structures  it  is  customary,  and 
also  highly  expedient,  to  throw  a  considerable  excess  of 
strength  in  favor  of  stability,  and  many  practical  men  have 
even  repudiated  theory  altogether,  as  leading  to  results  which 
cannot  be  relied  upon.  The  fault  has  been,  either  that  the 
theory  itself  was  erroneous,  or  sufficient  allowance  was  not 
made  for  imperfections  of  material  and  workmanship. 

With  a  theory  confirmed  by  experience,  and  with  resisting 
powers  assigned  to  the  materials,  sufficiently  far  below  the 
limits  given  by  experiments  on  perfect  specimens,  the  utmost 
confidence  can  be  placed  upon  the  results. 

Dimensions  arbitrarily  assumed,  in  accordance  with  the 
usual  custom,  are  certainly  less  to  be  relied  upon  than  those 
determined  upon  correct  principles  of  calculation. 

In  determining  the  weights  of  bridges,  it  is  necessary  to 
prepare  a  bill  of  timber  from  assumed  dimensions,  and  multi 
ply  the  number  of  cubic  feet,  by  the  weight  per  cubic  feet  of 
the  material ;  which  we  will  take,  as  an  average,  at  35  pounds. 
The  quantity  of  timber  will  be  assumed  (in  the  following  cal- 


APPLICATION   OF   RESULTS.  115 

dilations)  at  30  cubic  feet  per  foot  lineal,  as  this  is  about  an 
average  of  the  Howe  bridges,  on  the  Pennsylvania  Railroad. 
The  greatest  load  that  can  ever  be  thrown  upon  a  railroad 
bridge,  would  consist  of  several  locomotives,  of  the  first  class, 
attached  together  —  as  is  sometimes  done  in  clearing  off  snow  in 
winter. 

The  heaviest  locomotives  in  use  weigh  about  23  tons,  and 
their  length  is  23  feet.  Consequently,  1  ton  per  foot  for  the 
load,  and  -J  ton  per  foot  for  the  weight  of  the  structure,  may 
be  assumed  as  a  safe  average  for  the  maximum  load,  where 
the  span  does  not  exceed  200  feet.  One  and  a-half  tons  per 
foot  lineal,  will,  therefore,  be  assumed  as  the  extreme  load  — 
in  the  following  calculations. 

The  greatest  safe  strain  per  square  inch  for  wood,  will  be 
considered  as  1,000  pounds,  and  for  iron,  as  10,000  pounds. 

To  determine  the  strain  upon  the  chords. 

The  strain  upon  the  upper  chords,  is  one  of  compression  ; 
it  is  greatest  in  the  middle  of  the  bridge,  and  diminishes  to 
wards  the  ends.  The  maximum  strain  in  the  middle,  is  equal 
to  that  force  which,  if  applied  horizontally,  would  sustain  one- 
half  the  bridge,  if  the  other  half  were  supposed  to  be  removed. 
To  obtain  it,  multiply  half  the  weight  of  the  bridge  by  the 
distance  of  the  centre  of  gravity  from  the  abutment  (which  is 
always  very  nearly  one-fourth  the  span),  and  divide  the  pro 
duct  by  the  height  of  the  truss,  as  measured  from  the  middle 
of  one  chord  to  the  middle  point  of  the  other. 

Let  H  represent  the  horizontal  strain  in  the  centre 
S         "          "    span  of  the  bridge 
li  "    height  of  truss,  from  middle  points  of 

chords 

W        "          "    weight  of  the  whole  span 
W      8      1       SW 


Example. 

If  the  span  of  a  bridge  be  160  feet,  and  the  height  of  truss 
17  feet,  what  should  be  the  cross-section  of  the  upper  chord  in 
the  centre  ? 


116  BRIDGE   CONSTRUCTION. 

The  weight,  calculated  at  1J  tons  per  foot  lineal,  will  be 
480,000  pounds.  If  the  chords  be  12  inches  deep,  and  17  feet 
be  taken  as  the  measurement,  from  out  to  out,  the  distance 
from  the  middle  of  upper  to  the  middle  of  lower  chord  will  be 
16  feet ;  applying  the  formula,  we  have 

480,000  x  160 
H= 2o-rri7j =  600,000  pounds. 

o   X    J.D 

The  cross-section  of  the  chord,  to  resist  this  compression,  at 
1000  pounds  per  square  inch,  will  be  600  square  inches ;  and 
as  the  depth  is  12  inches,  the  total  breadth  must  be  50  inches; 
or,  25  inches  to  each  truss,  if  there  are  two  trusses. 

Of  the  strain  upon  the  lower  chord,  at  the  centre. 

The  strain  on  the  lower  chord  is  equal  in  degree  to  that  of 
the  upper,  but  it  is  tensile,  while  the  former  is  compressive. 

If  the  chord  could  be  made  in  a  continuous  piece,  without 
joints,  the  dimensions  would  not  be  required  greater  than  in 
the  former  case ;  but,  as  there  "is  generally  one  joint  in  every 
panel,  it  becomes  necessary  to  increase  the  quantity  of  mate 
rial  to  such  an  extent,  that  the  resisting  area,  exclusive  of  the 
joint,  shall  be  sufficient  to  resist  the  strain. 

This  requires,  in  general,  that  one  additional  line  of  chord 
timbers  should  be  introduced.  It  is  a  good  practical  rule  (and 
one  which  is  observed  in  Howe's  bridges),  to  make  the  upper 
chord  consist  of  three,  and  the  lower  of  four  timbers  to  each 
truss ;  a  joint  will  then  occur  in  each  panel,  and  the  pieces 
should  be  sufficiently  long  to  extend  over  four  panels.  With 
this  arrangement,  three  of  the  timbers  must  be  allowed  to  sus 
tain  the  whole  strain,  since  that  which  contains  the  joint  is  not 
capable  of  opposing  any  resistance. 

Strain  at  the  ends  of  the  chords. 

In  a  beam  resting  on  two  supports,  the  strain  at  the  end  is 
nothing,  and  increases  uniformly  to  the  centre,  but  in  a  bridge- 
truss  of  a  single  span,  there  will  be  a  horizontal  strain  at  the 
end  of  the  brace,  nearest  the  abutment,  which  will  equal  the 
weight  on  the  brace  multiplied  by  the  co-tangent  of  the  in- 


APPLICATION   OF   RESULTS.  117 

clination  of  the  brace.  If  the  inclination  of  the  brace  is  45°, 
the  horizontal  strain  will  be' equal  to  the  vertical  weight  upon 
A.  If  (as  is  generally  the  case)  the  angle  with  the  horizontal 
is  greater  than  45°,  the  horizontal  strain  will  be  less  than  the 
weight,  and  consequently,  it  will  be  safe  in  practice  to  assume 
the  horizontal  strain  at  the  end  of  the  chord,  or  more  correctly 
at  the  end  of  the  first  brace,  as  equal  to  the  vertical  force  act 
ing  on  that  brace. 

This  vertical  force  is  one-half  the  whole  weight  of  the 
bridge,  and  if  we  continue  the  calculation  with  the  dimensions 
already  given,  the  half  weight  will  be  240,000  pounds,  and  the 
cross-section  to  resist  it  240  square  inches,  or  a  little  more  than 
one-third  the  size,  required  in  the  centre. 

Having  determined  the  cross-section  of  the  chords  at  the 
centre  and  end,  a  uniform  increase  between  these  points  will 
fulfil  all  the  necessary  conditions. 

Another  circumstance  must  be  taken  into  consideration,  in 
determining  the  size  of  the  chords.  The  applied  weight  pro 
duces  a  cross-strain  upon  that  portion  of  the  chord  which  lies 
between  any  two  posts,  and  the  condition  of  the  chord  is  that 
of  a  beam  supported  at  the  ends,  and  loaded  in  the  middle. 

The  formula  is  R  —  9  ,   ,2,  or,  as  (b)  is  the  quantity  to  be  de- 

*j  0  CL 

tcrmined  b  =  9  ,2  ^.    If  the  interval  of  one  panel  be  assumed 

as  12  feet,  (1)  expressed  in  inches,  will  be  72,  d  =  12,  R  = 
1,000,  w  =  weight  in  an  interval  of  12  feet,  which  cannot 
exceed  6  tons  applied  at  the  centre.  By  substitution,  we  obtain 
3  x  12,000  x  72 

=  2x  1,000  x  12  »=   9mcheS' 

Hence  it  appears  that  the  size  of  the  chord,  as  determined 
by  this  condition,  is  much  less  than  by  the  former,  and  con 
sequently,  the  dimensions  previously  given  are  ample  to  resist 
the  cross-strain  arising  from  the  passing  load. 

Of  the  strain  upon  the  ties  and  braces. 

These  strains  are  always  estimated  together,  because  they 
bear  to  each  other  at  every  point,  the  proportion  of  the  height 


118  BKIDGE   CONSTRUCTION. 

of  a  panel  of  the  truss  to  the  diagonal ;  so  that  if  one  is  known, 
the  other  is  readily  determined  by  a  simple  proportion. 

In  a  simple  truss,  consisting  of  chords,  ties,  and  braces,  the 
braces  which  project  from  the  abutments  sustain  the  whole 
load.  The  weight  is  not  distributed  equally  amongst  all  the 
braces,  as  one  unacquainted  with  the  action  of  the  system 
might  suppose.  The  proportional  strains  on  each  successive 
brace,  from  the  centre  to  the  ends,  may  be  illustrated  by  a 
chain  suspended  from  a  fixed  point ;  the  upper  link  sustains 
the  whole  weight,  the  lower  none;  each  link  transmits  the 
weight  of  those  below  to  the  one  above  it ;  and  similarly,  each 
brace  transmits  the  strain  from  the  middle  of  the  span  to  the 
end,  adding  to  it  the  portion  due  to  the  panel  of  which  it  forms 
a  part.  The  end  braces,  unless  relieved  by  an  arch,  sustain 
the  whole  weight  of  the  structure,  and  its  load. 

As  the  weight  of  the  bridge  under  consideration  is  480,000 
pounds,  each  end  must  sustain  240,000  pounds,  arid  at  1,000 
pounds  per  square  inch ;  in  the  cross-section,  the  ties  must  be 
240  square  inches,  if  of  wood,  and  24  square  inches,  if  of  iron, 
allowing  in  the  latter  case  10,000  pounds  per  square,  inch  as  a 
safe  load,  although  in  practice  it  is  sometimes  greatly  exceeded. 

If  the  panels  be  12  feet  wide,  and  16  feet  high  in  the  clear, 
the  diagonal  or  brace  will  be  20  feet,  and  the  strain  on  the 

949  000  x  90 
brace  will  be  -      -jg—     -  =  300,000 ;  thus  requiring  300 

square  inches  of  wood,  or  4  braces,  of  75  square  inches  each. 

The  strain,  exactly  at  the  middle  point  with  a  uniform  load, 
is  theoretically  nothing ;  and  it  increases  from  this  point  to  the 
end,  where  it  attains  a  maximum  equal  to  one-half  the  whole 
weight  of  the  bridge ;  but  in  practice  there  never  is  a  brace 
exactly  at  the  middle ;  the  panels  must  have  considerable 
magnitude  in  a  horizontal  direction,  and  the  proper  estimate 
of  the  strain  is  that  which  would  be  produced  by  half  the 
maximum  weight  on  two  adjacent  panels,  or  the  whole  weight 
on  one  panel.  This  weight,  in  an  interval  of  12  feet,  will  be 
18  tons,  or  36,000  pounds ;  requiring  a  cross-section  of  only 
36  inches,  or  9  inches  to  a  tie,  and  11 J  to  a  brace. 

In  this  case,  the  cross-section  of  the  brace,  as  given  by  the 
condition  that  the  strain  .shall  not  exceed  1,000  pounds  per 


APPLICATION   OF   RESULTS. 

square  inch,  is  too  small,  since  its  length  would  cause  it  to 
yield  by  lateral  flexure.  To  obtain  the  proper  dimensions 
for  the  brace,  in  this  case,  "we  must  have  recourse  to  the  for 
mula  for  long  posts  ;  which,  for  white  pine,  gives  w  = 
9,000  b  d3 


If  we  assume  that  the  brace  is  20  feet  long,  without  inter 
mediate  support,  and  that  the  depth  is  5  inches,  the  breadth 

wl2 
will  be,  b  =          -      =  4. 


The  condition  which  has  been  assumed  is  not  a  common 
one  ;  the  braces  are  almost  always  supported  in  the  middle, 
which  reduces  the  length  of  the  unsupported  portion  to  one- 
half.  In  this  case,  the  formula  would  give  one-fourth  the 
breadth  required  in  the  former  case.  Very  small  braces 
would,  therefore,  be  sufficient  at  the  middle  of  the  bridge,  if 
supported  at  the  middle  of  their  length. 

It  was  calculated  that  each  of  the  four  braces  in  the  end 
panels  required  75  square  inches  of  cross-section  to  resist  the 
strain,  allowing  1,000  pounds  per  square  inch.  It  is  possible 
that,  even  with  dimensions  sufficient  to  furnish  this  area,  they 
may  yield  by  lateral  flexure.  To  test  this,  assume  the  depth 
as  9  inches,  and  breadth  8J  inches,  and  substitute  in  the  ex 
pression  for  the  weight,  which  becomes  w  =  185,880  pounds  ; 
a  result  which  proves  that  the  flexure  is  impossible  with  the 
weight  and  dimensions  assumed,  and  that  the  pressure  in  the 
direction  of  the  brace  is  the  only  one  to  be  provided  for. 

Having  determined  the  dimensions  of  the  ties  and  braces, 
at  the  centre  and  ends  of  the  truss,  the  intermediate  timbers 
should  increase  by  regular  additions.  If  the  truss  contain  16 
panels,  the  section  of  the  middle  braces  containing  15  square 
inches,  and  the  extreme  brace  75  inches,  the  intermediate 
braces  should  be  in  regular  proportion,  thus:  15,  23f,  321, 
40?,  49?,  57  1,  66|,  75. 

In  a  truss  thus  proportioned,  the  middle  braces  contain 
only  one-fifth  the  material  of  the  end  braces,  and  the  ties 
should  be  in  the  same  proportion. 


120  BRIDGE   CONSTRUCTION. 

Counter-braces. 

The  conclusion  arrived  at,  in  considering  the  subject  of 
counter-braces,  was,  that  the  greatest  strain  upon  a  counter- 
brace  was  equivalent  to  that  produced  by  the  action  of  the 
greatest  variable  load  upon  a  brace  ;  it  will  consequently  be 
equal  to  the  strain  upon  the  braces  of  the  middle  panel ;  and 
if  each  panel  contains  two  braces,  and  one  counter-brace,  the 
size  of  the  latter  should  be  uniform,  and  equal  to  30  square 
inches  of  cross-section,  in  the  truss  under  consideration.  A 
counter-brace,  5x6  inches,  supported  in  the  middle,  would 
afford  the  requisite  proportion. 

Lateral  horizontal  braces. 

The  use  of  lateral  bracing  is  principally  to  guard  against 
the  effects  of  wind,  and  other  disturbing  causes,  tending  to 
produce  lateral  flexure  in  the  roadway.  The  ordinary  bracing 
to  resist  this  action  consists  of  ties  and  braces,  similarly  dis 
posed  to  those  in  the  main  truss,  except  that  equal  strength  is 
required  in  the  direction  of  each  diagonal  of  the  horizontal 
panels.  The  greatest  lateral  strain  is  that  produced  by  the 
action  of  a  high  wind ;  assuming  the  force  of  wind  at  15 
pounds  per  square  foot,  as  a  maximum,  and  allowing  the 
height  of  truss  to  be  18  feet,  the  uniform  weight  over  the 
surface,  if  weather-boarded,  would  be  43,200  pounds,  or  21,600 
pounds  to  each  series  of  braces,  top  and  bottom.  The  effect 
of  this  force  would  be  estimated,  precisely  as  the  strain  of  a 
uniform  load  upon  a  bridge,  and  if  the  angles  of  the  lateral 
braces  be  45°,  the  diagonal  would  be  to  the  side  as  1.  4  :  1, 

21  600 
nearly.     The  strains  on  the  end  braces  will  be,  --^ —  x  1.  4 

M 

=  15.  120 ;  which,  at  1,000  pounds  per  square  inch,  would  re 
quire  but  ISy1^  square  inches  to  resist  the  strain;  or,  if  the 
brace  is  16  feet  long,  supported  in  the  middle,  and  depth  5 

iv  I2 

inches,  the  breadth  as  determined  by  the  formula  5  =  ?r7T7TA~P 

9,000  a3 

will  only  be  lT3o  nearly :  4  X  5  would,  therefore,  be  sufficient 
for  the  largest  lateral  brace  at  the  ends.  At  the  middle  of  the 


APPLICATION   OP   RESULTS.  121 

span  the  lateral  braces  would  be  exceedingly  light  ;  they  might 
even  be  omitted  in  the  middle  panel  without  injury. 

In  long  spans,  this  diminution  in  the  sizes  of  the  braces  in 
the  middle  adds  considerably  to  the  strength,  by  relieving  the 
bridge  of  unnecessary  weight. 

Diagonal  braces. 

These  timbers  occupy  the  direction  of  the  diagonals  of  the 
cross-section  of  the  bridge  ;  they  are  admissible  only  when 
the  roadway  is  on  the  top,  and  are  of  great  utility  in  prevent 
ing  side  motions  ;  where  the  roadway  is  on  the  bottom,  knee- 
braces  must  be  substituted. 

The  strains  upon  these  timbers,  which  result  from  unequal 
settling,  cannot  be  calculated,  as  it  is  impossible  to  determine 
the  side  to  which  the  bridge  may  have  a  tendency  to  settle  ; 
the  only  rule  is  to  make  them  large  enough. 

Experience  has  shown  that  5  X  6  is  sufficient  for  knee- 
braces,  and  5x7  for  diagonal  braces  in  oi-'linary  cases. 

Floor  beams. 

Allowing  the  unsupported  interval  between  the  trusses  to 
be  14  feet,  the  depth  of  beam  14  inches,  and  the  greatest  load 
equivalent  to  6  tons,  applied  at  the  centre  ;  required  the  breadth 
to  allow  a  deflection  of  j^th  inch  to  one  foot  :  timber,  white 

B  D3 

pine.    The  formula  for  this  case,  is  w  =  .AIO    72  wnence>  B  == 


wJ2x-0125    _  12,000  x  142x  -0125 

—jp-  "TP"  =      *  incnes- 

Having  now  completed  the  calculations  for  all  the  parts  of 
an  ordinary  truss,  composed  of  chords,  ties,  braces  and  counter- 
braces,  it  remains  to  estimate  the  effects  of  the  introduction  of 
arches,  and  the  combinations  of  different  systems  with  each 
other. 

In  entering  upon  the  consideration  of  this  subject,  it  is  pro 
per  to  remark,  that  our  calculations  must  be  based  to  some  ex 
tent  upon  uncertain  data  ;  for  where  two  systems  are  com 
bined,  we  cannot  be  certain  that  each  sustains  an  equal  por- 


122  BRIDGE   CONSTRUCTION. 

tion  of  the  weight ;  but,  on  the  other  hand,  we  are  sure  that 
the  assertion  sometimes  made,  that  either  one  or  the  other 
necessarily  sustains  the  whole  load,  is  erroneous.  Much  de 
pends  upon  the  manner  of  making  the  connection  ;  if  an  ordi 
nary  truss  be  constructed,  and  arches  added  after  it  has  settled 
to  a  considerable  extent,  by  the  application  of  heavy  weights, 
it  is  very  clear  that  the  arch  will  bear  but  a  small  proportion 
of  the  load ;  but  if  the  arch  is  introduced,  previous  to  the  re 
moval  of  the  false  works,  and  both  systems  be  allowed  to  set 
tle  together,  it  is  fair  to  suppose  that  the  strain  upon  each  will 
be  in  proportion  to  the  respective  power  of  resistance. 

The  usual  method  of  constructing  bridges,  is  to  make  the 
truss  of  such  strength  as  is  supposed  sufficient  to  support  the 
weight,  and  to  add  the  arch  as  additional  security.  We  think 
it  decidedly  preferable  to  reverse  this  arrangement,  making  the 
arch  the  main  dependence,  and  using  a  light  truss  in  combina 
tion  with  it,  merely  to  prevent  change  of  figure  in  the  arch, 
and  to  give  the  proper  elevation  or  inclination  to  the  roadway. 

Let  a  railroad  bridge  of  160  feet  span  be  supported  by 
four  arches,  the  rise  of  each  of  which  is  20  feet ;  weight  on 
bridge,  14  tons  per  lineal  foot — required  the  dimensions  of  the 
arches  to  sustain  the  whole  weight. 

The  whole  weight  is  240  tons,  or  240,000  pounds  to  each 
half  of  the  bridge.  The  strain  upon  the  arches  in  the  centre 

240,000x40 
will  therefore  be  -    — ^7) =  480,000   pounds,  requiring 

480  square  inches  of  cross-section,  at  1,000  pounds  per  square 
inch. 

Four  arches,  16  inches  deep  and  7^  inches  wide,  could 
supply  the  requisite  amount  of  material.  The  compression  at 
the  ends  will  be  to  that  in  the  centre  as  V402  +  20 2  :  40,  or 
as  v/2*+  1  :  2;  hence,  it  will  be  480,000  x  1|  nearly,  = 
540,000,  and  will  require  540  square  inches ;  or,  if  the  arches 
are  7^  inches  wide,  as  before,  the  depth  must  be  18  inches. 

If  an  arch  is  too  small  to  sustain  the  whole  weight,  and  is 
connected  with  a  truss  of  given  dimensions,  the  best  practical 
manner  of  treating  the  case  is  to  estimate  separately  what 
each  would  sustain,  allowing  1,000  pounds  per  square  inch ; 


APPLICATION   OF   RESULTS.  123 

and  divide  the  weights  in  proportion  to  the  powers  of  resist 
ance.  Having,  in  this  way,  determined  the  weight  to  be  sus 
tained  by  the  truss,  the  parts  can  be  proportioned  in  the  man 
ner  previously  explained. 

It  is  very  evident  that  an  arch  can  be  made  to  sustain  the 
whole  of  the  weight,  for  if  a  truss  has  settled  it  may  be  raised 
to  any  extent,  by  the  addition  of  arches  and  suspension  rods. 
In  this  case,  the  principle  of  proportioning  the  braces,  so  as  to 
increase  in  arithmetical  progression  from  the  middle  to  the 
end,  is  no  longer  applicable ;  there  is  no  more  strain  at  the 
ends  than  in  the  centre,  and  but  little  at  any  point,  and  in  this 
case  the  truss  is  of  no  other  use  than  to  stiffen  the  arch  and 
carry  the  roadway. 

Amount  of  counter-bracing  which  an  arch  requires. 

That  a  very  slight  force  is  sufficient  to  counter-brace  an 
arch,  may  be  rendered  evident,  without  a  calculation  in  detail, 
by  taking  a  more  unfavorable  case  than  could  possibly  occur 
in  practice.  Let  A  and  B  (Fig.  97)  represent  the  skew  backs 
of  an  arch,  and  leaving  out  of  consideration  the  resistance  of 
the  lower  chord,  which  adds  greatly  to  the  stiffness ;  suppose 
a  weight  of  1J  tons  per  foot  to  be  placed  on  one-half  of  the 
arch,  the  weight  of  the  other  half,  being  J  ton  per  foot,  will 
leave  1  ton  per  foot  to  produce  a  change  of  figure.  The  effect 
of  this  weight  will  be  represented,  nearly,  by  one-half  applied 
at  the  middle  point  (p).  Let  the  span,  s  =  160  feet ;  and  the 
rise  of  the  arch,  r  =  20  feet,  the  weight  at  p  =  (iv)  will  be 
40  tons.  Let  ^represent  the  component  of  the  weight,  in  the 
direction  of  the  chord  A  p.  At  the  centre,  the  value  of  this 

n  t j 

component  would  be  j^v/4r2  -f-  s2  =  82  tons,  and  as  it  is 

always  less  at  every  other  point,  the  slight  error  will  be  on  the 
safe  side,  by  taking  82  tons  as  the  force  acting  along  A  p. 
This  force,  and  its  equal  at  A,  gives  a  resultant,  acting  upwards 

at  m,  which  is  expressed  by  -  -  X  82  x  2.  In  the  present  ex 
ample,  o  n  =  12  and  o  p  =  60  nearly ;  hence,  the  force  at  m, 
which  acting  upward  must  be  resisted  by  the  counter-braces, 


124  BRIDGE    CONSTRUCTION. 

is  65  J  tons,  requiring  only  65  J  square  inches  of  resisting  area 
to  eacli  side  truss.  A  single  stick,  8x8,  would  therefore  be 
nearly  sufficient,  and  when  it  is  considered  that  the  strain  is 
not  all  at  one  point,  m^  as  we  have  supposed  it,  but  is  distri 
buted  over  a  considerable  length  of  arc,  the  amount  of  counter- 
bracing  necessary  to  resist  it  must  be  very  small. 

The  principle  of  determining  the  size  of  the  counter-brace, 
by  the  force  that  would  be  required  to  resist  the  upward  ac 
tion  of  the  arch,  is  not  that  which  we  recommended  when 
this  subject  was  considered. 

It  is  preferable  to  make  them  sufficiently  strong  to  throw 
a  permanent  strain  upon  the  arch  equal  to  that  produced  by 
the  passage  of  the  load,  and  this  condition  requires  as  much 
resisting  surface  as  that  presented  by  the  middle  braces.  It 
is  unnecessary  to  continue  the  application  of  these  principles 
to  a  greater  extent ;  we  believe  that  every  case  of  much  prac 
tical  importance  has  been  considered ;  and  the  illustrations 
given  will  be  sufficient  to  indicate  the  manner  in  which  the 
results  obtained  can  be  applied  to  the  determination  of  the 
dimensions  of  other  structures.  We  propose,  in  the  second 
part  of  this  work,  which  will  be  devoted  to  an  examination 
of  particular  modes  of  construction,  to  enter  more  into  detail, 
when  an  opportunity  will  be  offered  of  supplying  any  defi 
ciencies  that  may  exist,  and  of  illustrating  the  modes  of  calcu 
lation  by  which  the  strains  may  be  determined,  and  the  parts 
proportioned,  in  every  variety  of  combination. 


EQUILIBRIUM  OF  ARCHES. 


WE  cannot,  perhaps,  introduce  this  subject  better,  or  express 
our  own  views  of  it  more  clearly,  than  by  presenting  the  reader 
with  the  following  brief  exposition  of  the  ordinary  mode  of 
investigation,  as  copied  from  a  manuscript  procured  from  a 
brother  engineer.  It  exhibits  no  new  principle,  and  the  for 
mulas  are  deduced  in  a  similar  manner  fo  that  which  has  been 
used  in  Hutton's  Mathematics ;  but  as  the  facts  which  it  con 
tains  are  important,  and  must  form  the  basis  of  every  correct 
theory  for  determining  the  conditions  of  equilibrium,  we  will 
give  the  explanation  of  this  method,  and  then  proceed  to  point 
out  what  we  believe  to  be  its  defects. 

.fKOBLEM. 

To  find  the  thickness  of  abutments  of  arches,  of  any  kind. 

From  Gauthey  (modified). 

"  It  has  been  found  by  Monsieur  Boistard,  who  built  some 
good  bridges,  that  there  were  certain  points  in  an  arch  that 
were  weaker  than  others,  which  give  way  at  the  moment 
when  it  fails.  These  points  are  denominated  by  him  the 
points  of  rupture,  and  are  very  necessary  to  a  proper  solution 
of  this  interesting  problem,  which  is  now  very  much  simpli 
fied  by  the  author  above  named."  Gauthey  took  up  the  ex 
periments  of  Boistard,  and  upon  them  has  founded  the 
ing  solution. 

(125) 


126 


BRIDGE   CONSTRUCTION. 


Case  1st. 
FIG.  73. 


JLet  0  V  C'  be  the  intrados  of  any  arch,  whether  semi 
circular,  elliptical,  Gothic,  or  composite.  Let  D  be  the  crown 
of  the  extrados,  or  back  of  the  arch,  which  is  supposed  to  be 
filled  up  level  with  the  haunches  at  m  and  mr.  If  a  weight 
be  placed  upon  the  crown  too  great  for  it  to  bear,  it  yields,  and 
the  arch-stones  open  beneath,  at  the  crown,  while  the  extrados 
is  found  to  open  at  seme  point  on  each  side ;  either  at  the 
spring,  if  it  be  a  flat  arc  of  a  circle,  or  about  30  degrees  of  a 
semicircle,  or  at  various  other  points  if  it  be  composed  of  arcs 
of  circles,  tangent  to  each  other,  and  of  various  rises,  whether 
J,  or  J  or  I  of  the  span,  and  the  arch  only  falls  by  pushing 
aside  the  abutments  at  0  and  C',  the  opening  at  R  extending 
itself  up  to  the  top  at  m  and  m' '.  The  parts  of  the  arch  com 
prehended  between  the  joints  of  rupture  are  called  acting,  and 
the  rest  resisting.  It  has,  moreover,  been  observed  that  when 
the  abutment  gives  way,  it  leaves  a  portion  of  itself  standing, 
viz.,  XKs  ;  the  line  XK  being  at  an  angle  of  45°  with  the 
horizon,  which  only  adheres  by  the  strength  of  the  mortar  or 
cement  made  use  of. 

These  facts  being  stated  as  above,  we  may  now  consider 
the  manner  in  which  the  upper  part  acts  to  overturn  the  abut 
ments,  and  how  they  resist  that  action. 

Let  the  weight  of  the  portion  OR  m  P,  on  one  side  of  the 
crown,  be  represented  by  w,  this  weight  may  be  conceived  as 
supported  by  two  points  0  and  _Z),  and  pressing  upon  them  in- 


EQUILIBRIUM   OF   ARCHES.  127 

versely,  as  their  distance  from  the  vertical  line  passing  through 
the  centre  of  gravity  of  that  portion.  If  that  vertical  cut  C  Cr 
at  #,  and  we  call  0  Q,  a,  and  0  6r,  6,  then  the  whole  weight 
is  to  the  part  resting  on  (7,  as  b  :  b  —  a,  and  the  whole  weight 
to  the  pressure  at  D,  as  b  :  a.  Now  the  pressure  at  0  acts 
merely  to  keep  down  the  abutment,  and  that  by  a  leverage 
L  0 ;  but  the  pressure  at  D  produces  a  different  effect,  and 
one  that  must  be  carefully  attended  to.  Draw  Q  D  and  D 
a,  and  call  D  G,  c,  and  C  D,  d,  for  brevity.  The  two  half- 
arches  press  together  at  D,  and  mutually  sustain  each  other. 
Let  the  pressure  on  one  side  be  represented  by  D  Cr  (c),  and 
let  it  be  considered  separately,  and  apart  from  the  other  side, 
(<?)  may  be  decomposed  into  an  oblique  thrust  (c?),  and  a  hori 
zontal  action  (6),  which  last  acts  towards  6r,  and  tends  to 
crush  the  stones  at  the  key,  and  is  met  and  resisted  by  the 
strength  of  the  stone,  strongly  confined  between  the  pressure 
(b)  and  jts  equal  and  opposite  pressure  (6'),  so  that  we  have 
only  to  consider  the  oblique  action  (d)  which  evidently  bears 
from  (D)  towards  (0),  and  partly  tends  to  press  (C)  horizon 
tally,  and  partly  to  keep  it  down  vertically,  and  this  is  to  be 
added  in  part  to  the  resisting  forces ;  and  in  proportion  as  (Od) 
is  more  nearly  horizontal,  so  much  the  more  powerfully  it 
presses  (0)  horizontally,  and  vice-versa;  as  it  is  more  vertical 
the  more  does  it  tend,  as  in  Gothic  arches,  to  weigh  down  the 
abutments  and  keep  them  steady.  It  is,  moreover,  the  oblique 
pressure  which  this  part  of  the  arch  exercises,  which  squeezes 
the  arch-stones  so  tight  together  between  the  crown  (D)  and 
the  point  of  rupture  ((7),  as  to  make  them  act  as  one  homoge 
neous  mass,  or  stone,  whose  individual  parts  cannot  slip  out, 
even  though  they  should  not  be  shaped  as  wedges. 

The  former  notion,  about  the  arch  being  perfectly  equili- 
briated  by  a  catenarian  curve,  is  now  regarded  as  a  fallacy,* 

-"  We  agree  with  the  writer,  when  the  catenarian  curve  is  taken  as  the 
intrados,  but  when  it  is  used  to  determine  the  direction  of  the  joints,  and 
the  latter  are  made  perpendicular  to  it,  we  regard  it  as  any  thing  but  a 
fallacy.  With  as  much  propriety  might  the  practice  of  building  a  vertical 
wall  with  horizontal  courses,  that  is,  with  beds  perpendicular  to  the  line 
of  direction  of  the  pressures,  be  regarded  as  a  fallacy 


128  BRIDGE   CONSTRUCTION. 

and  the  whole  matter  at  present  rests  upon  the  relative  degrees 
of  action  of  the  upper  and  lower  parts,  or  the  parts  above  and 
below  the  points  of  fracture.  It  may  be  proper  here  to  remark, 
that  any  additional  weight  upon  the  crown,  such  as  is  often 
seen  in  heavy  banks  over  culverts,  may  be  easily  taken  into 
consideration,  as  all  that  part  which  rests  vertically  over  the 
acting  portion,  tends,  through  their  common  centre  of  gravity, 
to  produce  similar  results  to  the  masonry  itself,  and  all  the 
additional  weight,  which  is  just  over  the  resisting  parts,  has 
the  eifect  of  keeping  the  abutment  in  its  place.  We  should 
not  regard  the  pressure  of  earth  behind  the  walls  (although 
this  has  undoubtedly  a  very  great  effect  in  preserving  their 
stability),  because,  by  some  flood  of  the  stream  or  canal  the 
embankment  may  be  washed  away,  and  then  if  the  abutments 
had  not  been  calculated  to  sustain  the  pressure  of  the  arch, 
they  will  be  overthrown.  Besides,  earth  is  very  compressible, 
and  the  abutments,  although  sustained  from  actually  falling 
by  the  pressure  of  the  bank  behind  them,  may  yield  a  little, 
and  thus  disfigure  the  work. 

Having  thus  premised  the  general  considerations,  we  are 
now  prepared  to  go  into  the  algebraic  forms  for  expressing  the 
exact  quantities  of  each  of  the  acting  and  resisting  forces. 

First. — For  the  portion  of  (w\  resting  on  (O)  and  (D), 

b  :  a  ::  tv  :  -y-,  the  weight  resting  on  D, 

w 
b  :  b  —  a  : :  w  :  -7-  (b  —  a),  the  weight  resting  on  0 —  (A). 

Let  the  weight  on  D  be  decomposed  into  the  oblique  force 
acting  in  (d)9  and  the  horizontal  one  in  (Z»),  thus, 

w  a    wad 

c :  d  ::  —r-  :  -7 — ,  for  the  oblique  force  acting  from  D  to 
wards  (7, 

w  a    wad 
cio'.:  —j-  :  —r — ,  for  the  horizontal  crushing  force,  acting 

from  0  towards  (7.     This  last  is  neutralized  by  its  opposite, 

a  a. 

The  oblique  force  from  D  towards  (7,  is  now  to  be  decom 
posed  into  two  others,  the  one  acting  downwards  at  (7,  the 
other  acting  horizontally  from  6r  towards  (7. 


APPLICATION  OF    RESULTS.  129 

w  a  d    iv  a 
a  :  o  as  --T —  :  —,—  which    is    nothing    more    than    to    say 

+4  Q  C  0 

that  the  weight  on  D  is  transferred  by  the  oblique  action  to  (7, 
for  it  is  the  very  same  as  the  first  expression  above  for  the 
weight  on  D;  add,  therefore,  the  last  obtained  vertical  force 
on  0  to  the  second  expression  (A),  showing  the  weight  on 
(7,  and  we  have 

Wa       7F/7  w    ,  wb 

_+_.(5_a)==j(J  +  a_fl)  =  _=> 

showing  that  however  the  forces  have  been  considered  under 
their  various  actions,  they  still  result  in  simply  resting  one- 
half  of  the  upper  portion  of  the  whole  arch  on  (7,  the  other 
half  being  borne  by  <?;,  which  is  obvious.  Next,  to  obtain  the 
horizontal  thrust  produced  by  said  oblique  force  acting  inD  (7, 

w  a  d      w  a 

Ct  I  u  I  '        j  I 

be  c 

This  last  is  the  only  force,  which  has  any  effect  to  overturn 
the  arch,  and  acts  by  its  leverage  L  X,  which  we  may  call 
(e),  X  being  the  fulcrum  or  pivot  around  which  it  would  turn 

w  a  e 
m  its  overthrow ; is  then  the  moment  of  the  acting  force. 

W,  the  weight  of  half  the  upper  part,  acts  at  0  at  the-  end 
of  the  lever  X  S  which  we  shall  call  (/),  to  keep  the  arch  in 
place.  If  (u)  be  the  weight  of  all  the  resisting  portions,  which 
may  properly  be  called  abutments,  and  a  vertical  be  passed 
through  its  centre  of  gravity,  falling  at  a  distance  (g)  from 
X,  then  (u)  has  (g)  for  its  lever,  and  (u  g)  is  the  effect  of  the 
abutment  to  resist  the  action,  to  which  we  must  add  wf,  for 
the  sum  of  all  the  resisting  forces,  thus,  u  g-  +  wf. 

T    .  iv  a  e 

It  is  needless  to  say  that  u  g  -f  wj  =  when    m  equi- 

0 

librium.  If  the  lower  part  is  built  with  the  best  hydraulic 
cement,  it  may  be,  that  its  cohesive  force  on  the  joints  of  frac 
ture  X K,  will  keep  the  triangle  X K S  from  separating  iia 
its  overthrow ;  this  will  evidently  depend  on  the  quality  of  the 
cement,  and  would  lead  us  to  conclude  that  no  expense  should 
be  spared  to  make  the  mortar  of  the  abutments-  as  good  as 
possible  in  all  cases. 
9 


130  BRIDGE    CONSTRUCTION. 

It  may  be  useful  here  to  adduce  by  way  of  practical  illus 
tration  of  the  above  theory,  a  case  which  did  actually  occur,  in 
the  year  1829  or  '30,  in  constructing  the  Chesapeake  and  Ohio 
Canal.  The  Monocacy,  a  very  violent  stream,  is  crossed  by  a 
beautiful  stone  bridge  (aqueduct),  of  9  arches,  each  54  feet 
span,  and  9  feet  rise ;  arches  2J  feet  thick,  abutments  10  feet 
thick,  and  10  feet  high  on  a  foundation  of  3  feet  high,  and  13 
feet  wide. 

Some  arches  and  piers  had  been  built  up  and  backed  in, 
but,  before  the  whole  could  be  completed,  a  great  flood  swept 
away  the  last  centre  from  under  the  arch  just  turned  and  not 
backed  in,  except  partially  on  one  side :  the  rise  of  this  arch 
being  only  one-sixth  part  of  the  span,  must  have  pressed  with 
tremendous  effect  upon  its  last  pier,  especially  as  the  supports 
were  very  suddenly  knocked  from  beneath  it,  and  it  was  brought 
to  bear  very  suddenly  upon  the  pier.  This  had  been  well 
built  with  hydraulic  cement  of  tolerably  good  quality,  only 
eight  or  ten  months  before.  The  arch  stood  triumphantly,  and 
contrary  to  the  expectation  of  all  that  witnessed  it,  who  looked 
for  nothing  but  the  destruction  of  every  arch  then  built  one 
after  another.  But,  upon  a  subsequent  investigation  of  the 
thrust  and  resistance, 

the  former  —      —  was  found  to  be  126,300  pounds, 

the  latter  u  y  +  wf  was  found  to  be  162,390  pounds, 
showing  a  considerable  surplus  of  strength.  Its  standing  was 
then  attributed  to  the  great  strength  of  the  cement,  and  excel 
lence  of  workmanship;  but  the  cement  having  been  since  tried, 
and  found  to  slake  in  the  air  like  common  lime,  it  was  no  bet 
ter  than  good  lime-masonry  should  always  be,  and  its  standing 
must  be  attributed  to  intrinsic  weight  and  strength. 

a  =  12-5         9  =  5- 

I  =  27.  w=  11,620  pounds 

c  —  11-5         u  —    9,438  weight  of  a  cubic  foot  of  stone, 

assumed  —  140  pounds 

e  =  10- 

f  =  10*  w     having  lost  much  of  its  specific  gravity 

by  immersion. 


APPLICATION   OF   RESULTS.  131 

Case  2d. 

There  is  another  point  of  view,  under  which  the  yielding 
of  an  arch  may  be  considered.  The  upper  portion  acting  as 
above  mentioned,  and  the  lower  parts,  or  abutment,  being 
sufficient  to  resist  them  and  to  keep  from  being  overturned, 
the  arch  may  give  way  from  the  breaking  up  and  sliding  of 
the  stones  one  upon  another,  at  some  horizontal  joint ;  suppose 
for  instance,  at  L  0:  in  that  case,  the  resistance  must  depend 
entirely  on  friction,  and  on  the  strength  of  the  mortar. 

Boistard  has  found,  from  numerous  experiments  on  the  sub 
ject,  that  friction  is  0-76  w  (w  being  the  weight  resting  over 
the  joints  of  rupture),  and  that  the  strength  of  adhesion  of  the 
mortar  is  3,900  pounds  per  square  foot.  Now  assuming  this 
as  true  for  a  majority  of  cases,  though  evidently  subject  to 
much  modification,  and  dispensing  with  the  leverage  used  in 

Wa 
the  first  case,  we  shall  have  the  horizontal  thrust  simply . 

And  calling  the  strength  of  mortar  per  square  foot  (s),  and  the 
number  of  square  feet  area  of  mortar  joint  (h\  (u)  the  superin 
cumbent  weight  above  the  joint,  and  (?*)  the  friction  of  the 
sliding  parts,  the  resistance  will  be  (W+u)r  +  sh;  and 

Wa 
when  on  exact  balance,  -    -  —  ( W  -f  u)  r  +  s  h. 

That  is,  the  sliding  joint  (assuming  several  for  supposition), 
where  the  resistance  is  found  least  in  relation  to  the  sliding 
force  opposed  to  it,  is  generally  at  the  springing  line,  for  above 
that  the  area  of  joint  increases  rapidly,  and  below  it  the  weight 
causes  more  friction. 

This  leads  to  the  practical  consideration  which  has  made 
eminent  bridge  builders  change  their  former  practice  very 
much,  that  the  more  uneven  and  projecting  the  stones  in  the 
abutments,  near  the  springing  line,  and  the  more  inclined  to 
wards  the  thrusting  line,  the  more  effectually  will  they  resist. 
For  this  reason  arches  are  sometimes  continued  through  the 
abutments ;  at  other  times,  stones  are  set  up  at  frequent  inter 
vals  on  end,  amongst  the  others,  and  the  masons  are  forbidden 
to  course  behind. 


132  BRIDGE   CONSTRUCTION. 

Gauthy  found  that,  in  some  cases  compared  by  him,  it  re 
quires  a  greater  thickness  of  abutment  for  the  second  case  than 
for  the  first. 

Thickness  of  Pos.  of  joint  Thickness  of    Pos.  of  joint 

abutments.  of  fracture,  abutments.      of  fracture. 

For  semicircles                                       1'47  ft.              30°  4'31  ft.  15°. 

anse-paniers  rising  ^  the  span  2-16  ft.             50°  5-30  ft.  35°. 

anse-paniers  rising  i  the  span  2-68  ft.             60°  7'32  ft.  45°. 

The  arch  above  supposed,  is  67*4  feet  span,  arch-stones 
3-27  feet  long,  backed  up  level,  and  springing  from  the  broad 
platform  of  the  foundation  without  any  height  of  abutment. 

But  where  the  abutments  have  height,  as  in  ordinary  cases 
of  smaller  arches,  the  thickness  found  by  him  would  have  been 
vastly  increased,  on  account  of  the  great  increase  of  thrust 
from  greater  leverage. 

The  actual  position  of  joints  of  fracture  can  only  be  found 
by  trial  of  several  suppositions,  and  that  is  to  be  taken  where 
the  resistance  is  weakest  when  compared  to  the  thrust  at  that 
point,  or  where  they  are  most  nearly  equal,  and  consequently 
their  ratio  is  the  least. 

It  is  well  to  remember  that  the  resistance  is  much  dimin 
ished  when  the  abutments  are  immersed  in  water,  as  in  piers 
in  rivers. 

In  the  case  of  the  Monocacy  aqueduct,  tried  upon  the  last 

supposition,  we  find =  12,630  pounds,  and  (W  +  u)  r  + 

8  h=  50-023  pounds,  showing  a  greater  excess  of  resistance 
than  in  the  other  case,  supposing  it  to  yield  by  overturning. 
If  in  an  arch  the  joints  of  rupture  be  at  the  springing  lines 

and  the  extrados  of  the  crown,  the  horizontal  thrust  is  -7-,  in 

which  a  is  the  distance  from  the  springing  line  to  the  perpen 
dicular,  through  the  centre  of  gravity ;  b  is  the  vertical  dis 
tance  equal  to  the  rise  of  arch  +  thickness  of  ringstones,  w  the 
weight  of  half  the  arch. 

The  conditions  of  equilibrium  of  the  abutment  are  simply 
that  the  moments  of  the  horizontal  and  vertical  forces  shall  be 
equal,  the  weight  of  the  arch  being  applied  at  the  springing 
Une. 


APPLICATION   OF   RESULTS. 


133 


Although  it  would  appear  from  the  preceding  statements, 
copied  literally  from  the  manuscript  referred  to,  that  the  results 
given  by  this  formula  can  be  relied  upon  in  practice,  yet,  not 
withstanding  the  evidence  furnished  by  the  Monocacy  aque 
duct,  we  cannot  think  that  the  dimensions  given  by  the  for 
mula  are  sufficient.  When  the  stone  is  extremely  hard,  and 
the  pressure  upon  it  very  small  in  proportion  to  its  capability 
of  resistance,  the  result  may  be  sufficiently  great,  but  in  other 
cases  it  cannot  be  trusted.  In  fact,  it  is  evident  that  the 
formula  has  been  deduced  upon  the  supposition  that  the  pres 
sures  are  thrown  entirely  upon  the  points  D  and  0,  but,  unless 
the  strength  of  the  material  be  almost  infinite,  these  points 
could  not  sustain  the  pressure ;  the  portions  of  the  stone  lying 
at  these  points  would  break  off,  and  the  points  of  contact  D 
and  (7,  being  thus  brought  nearer  together,  would  render  the 
line  of  direction  of  the  pressure  more  nearly  horizontal,  in 
crease  both  the  horizontal  force  at  0  and  the  leverage  c  s,  or 
L  X  at  which  it  acts,  and  consequently  require  a  greater  thick 
ness  of  abutment  to  resist  its  effects. 

That  which  we  believe  to  be  the  true  method  of  determin 
ing  the  equation  of  equilibrium  of  an  arch,  can  be  deduced 
from  a  process  of  reasoning  analogous  to  that  employed  in  the 
case  of  a  straight  beam  supported  at  the  ends,  or  the  chords  of 
a  straight  bridge. 

The  lower  fibres  of  a  beam,  and  the  lower  chords  of  a 
straight  bridge-truss  are  in  a  state  of  extension,  and  the  upper 
ones  of  compression,  and  the  neutral  axis  is,  in  general,  in  the 
middle  of  the  depth ;  but  in  an  arch  of  any  material,  resting 
upon  fixed  abutments,  the  resistance  of  the  abutments  exactly 
replaces  that  of  the  ties  or  lower  chords  in  the  former  case,  and 
the  position  of  the  neutral  axis  will  remain  unchanged. 

FIG.  74. 


134  BRIDGE    CONSTRUCTION. 

Let  D  B  represent  a  half  arch.  Draw  A  P,  0  or  and  D 
P.  If  0  P  =  A  B  the  resistance  of  the  abutments  acting  in 
the  direction  D  P  will  produce  the  same  effect  as  a  tie  in  the 
same  direction,  and  capable  of  opposing  the  same  resistance. 
Since,  therefore,  there  is  a  change  from  extension  at  P,  to 
compression  at  A,  there  must  exist,  as  in  beams  or  straight 
bridges,  a  neutral  axis  between  A  and  P  ;  and  as  A  B,  as  will 
be  shown,  equals  0  P,  the  neutral  axis  will  bisect  A  P. 

The  pressure  upon  any  given  point  of  the  joint  A  B,  will 
be  as  its  distance  from  the  neutral  axis ;  and  if  the  perpendic 
ular  A  n  represents  the  maximum  strain  upon  a  square  unit 
at  A,  join  On,  and  the  perpendicular  of  the  triangle  A  On  will 
represent  the  proportional  pressures  upon  other  points.  The 
whole  pressure  upon  the  joint  will  be  represented  by  the  trape- 
zoid,  B  n.  A  perpendicular  to  A  B,  through  the  centre  of 
gravity  of  the  trapezoid,  will  give  the  centre  of  pressure  of  the 
joint  A  B,  which,  when  OB  equals  or  exceeds  A  B,  or  in 
other  words,  when  the  rise  of  the  arch  is  greater  than  about 
three  or  four  times  the  depth  of  the  arch-stones,  will  be  suffi 
ciently  near  the  centre  of  the  joint  to  render  the  error  made  by 
taking  it  at  the  centre  very  small,  and  that  too  on  the  side  of 
stability. 

When  greater  accuracy  is  required,  the  centre  of  gravity  of 
the  trapezoid  must  be  found.  As  a  general  rule,  we  think  that 
practical  formulas  of  this  kind  should  be  made  as  simple  as 
possible,  and  that  instead  of  aiming  at  the  greatest  theoretical 
accuracy,  it  is  best  to  reject  small  errors  that  are  in  favor  of 
stability,  in  order  that  the  formula  may  give  an  excess  of 
strength.  As  an  illustration  of  the  little  reliance  that  practical 
men  place  upon  the  deductions  of  theory,  we  will  state,  that 
the  dimensions  assigned  to  parts  of  structures  are  often  twice 
as  great  as  the  rule  allows.  Such  a  difference  should  not 
exist;  the  dimensions  of  structures  deduced  from  theoretical 
considerations  should  correspond  with  those  assigned  in  prac 
tice,  and  in  order  that  this  may  be  the  case,  the  theory  must 
be  based  on  correct  principles,  and  include  every  circumstance 
which  tends  to  derange  the  stability. 

The  ordinary  equations  of  equilibrium  will  therefore  give 


APPLICATION   OF    RESULTS.  135 

results  sufficiently  near  the  truth,  by  taking  the  middle  of  the 
joints,  instead  of  the  points  A  and  D,  as  the  points  of  applica 
tion  of  the  pressures. 

It  is  very  necessary  to  observe  that  the  equation  of  equi 
}ibrium  above  determined  is  based  upon  such  conditions,  that 
the  resultant  of  all  the  forces,  both  of  the  acting  and  resisting 
portions,  passes  through  the  point  x  at  the  back  of  the  abut 
ment.  The  dimensions  thus  determined  will  be  sufficient 
only  in  the  case  of  a  rock,  or  other  incompressible  foundations  ; 
in  other  cases,  where  there  is  any  liability  to  yield,  the  result 
ant,  instead  of  passing  through  the  extremity  must  pass  through 
the  middle  of  the  base.  This  condition  is,  in  general,  best  ful 
filled  by  making  the  back  of  the  abutment  in  steps  or  offsets, 
which  permits  an  enlargement  of  the  base,  without  greatly  in* 
creasing  the  amount  of  masonry ;  and,  at  the  same  time,  favors 
stability,  by  throwing  the  centre  of  gravity  very  much  towards 
the  face. 

If  the  arch  between  D  and  B  were  in  one  solid  piece  with 
out  joints,  it  would  follow,  that  the  joints  A  B  and  C D,  being 
entirely  above  and  below  the  neutral  axis,  would  be  compressed 
throughout  their  whole  extent,  and  would  have  no  tendency 
to  open ;  but  cases  have  often  occurred  in  which  some  of  the 
joints  have  opened  at  the  back  or  front,  and  the  work  suffered 
considerable  derangement.  Such  an  effect  may  result  from 
two  distinct  causes. 

First. — When  an  arch  is  constructed  it  is  usual  to  commence 
by  laying  the  stones  nearest  the  abutment,  and  proceeding  to 
wards  the  centre ;  months  sometimes  elapse  between  the  lay 
ing  of  the  first  and  last  stones  of  the  arch,  during  which  time, 
if  the  cement  or  mortar  is  of  good  quality,  those  first  laid  be 
come  solidly  united  to  each  other.  If  the  centres  are  removed 
soon  after  the  completion  of  the  arch,  and  while  some  of  the 
joints  are  in  a  soft  or  compressible  state,  inequalities  of  settling 
must  result,  sufficient  in  some  cases  of  itself  to  account  for  all 
the  observed  derangement. 

The  second  case,  in  which  the  joints  of  an  arch  will  have 
a  tendency  to  open,  is  when  the  line  of  pressure  passes  below 
the  intrudes,  or  above  the  extrados.  To  guard  against  this 


136  BRIDGE    CONSTRUCTION. 

effect,  the  load  upon  the  different  parts  of  the  arch  and  the 
curve  of  its  intrados  must  bear  such  a  relation  to  each  other, 
that  the  line  of  pressure  will  never  fall  outside  the  limits  of 
any  joint,  but  will  approach  as  nearly  to  the  centre  of  the  joint 
as  possible. 

To  find  the  relative  length  of  the  joints  at  different  point* 
of  an  artih,  and  the  line  of  direction  of  the  pressure. 

FIG.  75. 


Let  c  d  represent  the  depth  of  the  joint  at  the  crown  neces 
sary  to  resist  the  horizontal  thrust,  as  determined  from  assumed 
dimensions,  and  let  this  force  be  represented  by  a  line  o  e., 
equal  to  c  d,  applied  at  the  centre  of  pressure  (o).  Let  6r  re 
present  the  centre  of  gravity  of  the  arch  A  d,  and  m  r  =  length 
of  line  that  represents  the  weight.  Transfer  the  force  at  o  to 
the  point  m,  and  make  m  ef  =  o  e.  Construct  the  parallelo 
gram  of  forces  m  8.  As  m  ef  represents  the  length  of  joint 
necessary  to  resist  the  horizontal  force,  m  r  would  be  the  length 
sufficient  to  sustain  the  weight,  and  the  resultant  m  s  would 
represent  the  length  of  a  joint,  to  resist  the  combined  pressure 
of  the  two  forces.  Draw  Ap  perpendicular  to  m  s,  produce 
and  equal  in  length  to  m  s.  Ap  will  represent  both  the  length 
of  the  joint  at  the  point  A,  and  its  proper  direction,  since  it  is 
perpendicular  to  the  line  of  pressure  m  s. 

By  drawing  p  n  parallel,  and  A  n  perpendicular  to  A  B, 
we  find  that  the  triangles  A  p  n  and  m  s  r  will  be  equal, 
hence,  An  =  sr  —  c  d,  and  as  the  same  is  true  at  any  other 
point  it  follows,  that  the  difference  of  level  of  the  extremities 
of  any  joint  of  the  arch  should  be  equal  to  the  depth  at  the 
?r:wn.  Also  as  p  n  =  m  r  —  weight  of  portion  of  arch  A  Dt 


APPLICATION   OF    RESULTS.  137 

t  follows,  that  the  Jwrizontal  distance  letiveen  the  extremities 
of  any  joint  will  be  proportional  to  the  weight  of  the  portion 
of  the  arch  betioeen  it  and  the  crown.  pf  being  the  point  of 
application  of  the  resultant  of  the  pressures  upon  all  parts  of 
the  joint  A  p,  and  p'  s  its  line  of  direction,  p'  s  must  be  tangent 
to  the  curve  of  equilibrium.  By  finding  the  point  p'  for  other 
joints  between  A  and  D,  the  curve  traced  through  them  will 
be  the  line  of  direction  of  the  pressures. 

The  manner  of  finding  the  point  pf  for  any  joint  A  p  is  ob 
vious;  it  is  the  intersection  of  the  line  A  p  with  the  diagonal  of 
the  rectangle,  one  of  whose  sides  ef  m  is  proportional  to  the 
horizontal  pressure,  and  is  constant  at  every  point  of  the  arch ; 
the  other,  m  r,  represents  the  weight  of  the  portion  A  d  of  the 
arch,  acting  through  Cr  its  centre  of  gravity.  The  position  of 
G-  can  be  readily  found  for  any  joint,  as  (u  u'}  by  making  a 
drawing  of  the  arch  on  pasteboard,  cutting  it  out  and  balanc 
ing  the  portion,  of  which  the  centre  of  gravity  is  to  be  ascer 
tained.  The  weight  can  .  be  found  either  by  weighing  the 
pasteboard,  or  by  calculation,  and  thus  we  are  furnished  with 
an  extremely  simple  and  practical  method  of  describing  the 
curve  of  equilibrium. 

The  method  usually  recommended  for  determining  practi 
cally  the  direction  of  this  curve,  is  to  mark  off  on  a  wall,  or 
other  vertical  surface,  the  span  and  rise  of  the  arch,  then  sus 
pend  a  flexible  chain  between  these  points,  and  load  it  at  short 
intervals  with  weights  proportional  to  the  superincumbent  por 
tions  of  the  arch.  As  the  addition  of  these  weights  will  change 
the  figure  of  the  curve,  the  length  of  the  chain  and  the  mag 
nitudes  and  distribution  of  the  weights  must  be  varied,  until 
by  successive  trials  the  proper  proportion  and  distribution  are 
discovered.  This,  which  is  recommended  as  a  very  simple 
method,  and  easy  of  application  by  any  practical  builder,  we 
conceive  to  be  exceedingly  troublesome,  and  such  as  no  practi 
cal  builder  would  be  likely  to  undertake ;  and  after  the  curve 
has  been  found  in  this  way,  we  know  nothing  of  the  position 
of  the  centres  of  pressure:  in  fact,  it  is  evident,  from  the  method 
which  has  been  pursued,  that  they  have  been  assumed  at  the 
springing  lines  and  at  the  lowest  points  of  the  key-joint,  as  these 


138 


BRIDGE   CONSTRUCTION. 


are  the  points  through  which  the  curve  has  been  drawn.  The 
method  which  we  have  ventured  to  recommend  determines 
the  curve  at  once,  without  the  necessity  of  successive  trials, 
and  also  gives  the  centres  of  pressure  of  every  joint. 

In  practice,  it  is  not  generally  necessary  to  find  the  curve 
of  equilibrium  and  trace  its  course  ;  its  principal  utility  is  in 
determining  the  direction  of  the  joints,  which  should  be  made 
perpendicular  to  it,  but  the  direction  and  length  of  the  joints 
can  be  readily  determined  without  it  as  follows. 

We  have  seen  that  the  horizontal  pressure  at  any  point  of 
an  arch  is  equal  to  that  at  the  centre,  and  is  constant ;  but  the 
vertical  pressure  is  variable,  and  equal  to  the  weight  of  the 
portion  of  the  arch  included  between  the  given  joint  and  the 
crown ;  therefore,  to  find  the  direction  of  any  joint,  as  u  u', 
draw  a  vertical  line  u  o'  at  the  point  u,  make  it  equal  to  c  d, 
the  depth  at  the  crown,  draw  o'  u'  perpendicular  to  it,  and 
bearing  to  it  the  same  proportion  that  the  weight  of  the  portion 
of  the  arch  u  d  bears  to  the  horizontal  thrust.  Join  u  uf, 
which  will  give  both  the  direction  and  length  of  the  joint. 

The  above  theory,  and  the  simple  rule  which  has  been  de 
duced  from  it,  may  be  verified  to  some  extent  by  applying  it 
to  the  case  of  an  arch  uniformly  loaded. 

Tredgold  and  others  have  shown  that  the  curve  of  equi 
librium  in  this  case  is  the  common  parabola,  and  the  proof  ia 
extremely  simple. 

FIG.  76. 


-A. 


Let  A  B  represent  a  portion  of  the  curve  supposed  to  bo 
uniformly  loaded.  Draw  B  n  parallel  and  Gr  o  perpendicular 
to  A  0.  The  weight  acting  at  the  centre  of  gravity  6r,  and 


APPLICATION    OF    RESULTS. 


13G 


the  horizontal  pressure  at  the  crown,  may  both  be  transferred 
to  n  (the  intersection  of  their  lines  of  direction).  The  resultant 
is  n  Ay  which  must  pass  through  A,  and  as  it  represents  the 
direction  as  well  as  the  intensity  of  the  pressure  at  A,  it  must 
be  tangent  to  the  curve. 

But  An  o  and  ADO  are  similar  triangles,  and  as  the 
weight  is  supposed  to  be  uniform,  and  as  a  consequence  A  o 
=  0  £,  it  follows  that  B  D  also  equals  B  (7,  which  is  a  well 
known  property  of  the  parabola. 

The  method  which  we  have  suggested  for  finding  the 
curve  of  equilibrium, -is  based  upon  the  principle  that  the  hori 
zontal  pressure  is  constant  at  all  points  of  the  arch,  and  the 
vertical  pressure  upon  any  joint  is  equal  to  the  weight  of  the 
portion  of  the  arch  between  that  joint  and  the  crown. 

FIG.  77. 


If  then  this  principle  be  correct  when  applied  to  the  parabola, 
it  follows  that  if  any  joint  be  taken,  as  6r,  and  a  line  drawn 
vertically  through  the  centre  of  gravity  of  Q-  B,  terminated  by 
the  line  drawn  horizontally  through  the  crown ;  if  n'  P  be 
made  to  bear  the  same  proportion  to  the  weight  of  Cr  B,  that 
B  R  does  to  the  whole  weight  on  A  B  or  B  0 ;  then  &P, 
which  represents  the  horizontal  component  of  the  pressure  at 
6r,  should  be  constant  at  every  part  of  the  curve,  and  be  equal 
to  A  W  or  i  A  R. 

To  prove  that  this  is  the  case,  and  that  the  parabola  con 
forms  to  the  rule  that  we  have  endeavored  to  establish. 

Take  B  R  to  represent  the  weight  on  A  R  and  call  it  x. 
Also  let  A  R  =  y.  Take  any  point  6r,  and  let  n  =  ratio  be 
tween  G-  u  and  A  R.  Gru  will  therefore  be  equal  to  (n  ?/), 


140 


BRIDGE   CONSTRUCTION. 


and  n'  P,  which  represents  the  weight  on  Gr  u,  will  be  equal 
to  nx.     From  the  equation  of  the  parabola  we  have 
y  —  <jpx  .'  .ny  =  n  Vp x,  or  since  Gru  —  ny—  ^p  x  B  u] 
we  will  have  ny  =  ^/px'  (by  calling  B  u  —  x1  for  brevity) .  *. 
p  xf  =  n2p  x  .  • .  xr  —  n2  x.     But  from  similar  triangles, 

n'O'i  0'  &::  n'P:  PS,  or  n2  x  :  -^  ::wz:«^=|  = 

AR 

— ^-  =  a  constant  quantity. 

NOTE. — The  fact  established  in  the  preceding  demonstration  furnishes 
a  convenient  method  of  describing  the  parabola  by  points. 

FlG.   78. 


•Zlt.  

^^ 

?T 

v/i 

&' 

or 

- 

Let  A  and  B  be  two  points  through  which  a  parabola  is  to  be  drawn. 
Divide  A  C  and  B  C  each  into  the  same  number  of  equal  spaces :  draw 
the  horizontal  and  vertical  lines  through  the  points  of  division  as  repre 
sented  in  figure.  Through  G  (the  middle  of  the  first  space)  draw  G  o  = 
Bn:  lay  off  om  —  iA  C:  draw  Gm,  and  its  intersection  (5)  with  the 
vertical  through  m  will  determine  a  point  of  the  curve,  the  apex  being 
at  B. 

Again,  on  the  vertical  through  m  (the  middle  of  m'  C)  lay  off  G o/  =2 
Bn  =  B  n'  make  o'  m'  =  i  A  C  as  before,  draw  G  mf  and  its  intersection 
with  the  vertical  through  ???/,  will  determine  s/ :  a  second  point  of  the 
curve. 

In  the  same  way  any  required  number  of  points,  at  e^ual  distances 
apart,  may  be  determined. 


ILLUSTRATIONS 


OF 


PARTICULAR  MODES  OF  CONSTRUCTION. 


As  the  object  in  this  treatise  is  to  explain  the  general  prin 
ciples  of  bridge  construction,  110  attempt  will  be  made  in  this 
first  part  to  enter  into  details,  but  as  the  subject  would  be  in 
complete  without  illustrations,  outlines  will  be  given  of  those 
structures  which  deserve  attention.  Some  new  combinations 
that  might  be  advantageously  employed  will  be  included. 


FOOT  BRIDGE  ACROSS  THE  RIVER  CLYDE. 

BY  PETER  NICHOLSON. 

FIG.  79. 


The  general  arrangement  of  the  supports  is  represented  in 
the  above  figure ;  it  consisted  of  piles,  driven  into  the  bed  of 

(141) 


142  BRIDGE    CONSTRUCTION. 

the  stream,  across  which  longitudinal  pieces  were  placed  to 
span  the  openings  ;  these  were  strengthened  by  a  framework 
on  top,  consisting  of  two  oblique  braces  with  a  straining-beam. 

The  same  kind  of  a  frame  is  much  used  at  present  for  span 
ning  short  intervals  ;  it  possesses  sufficient  vertical  strength, 
but  has  no  counter-bracing,  and  consequently  would  be  defi 
cient  in  stiffness.  For  a  foot-bridge,  particularly  one  which 
does  not  rest  upon  stone  supports,  its  flexibility  would  not  be 
a  serious  objection.  When  stone  supports  are  used,  every  pre 
caution  must  be  taken  to  prevent  vibration,  as  it  breaks  the 
mortar  of  the  joints,  loosens  the  stones,  and  rapidly  ruins  the 
structure. 

A  bridge  built  by  Palladio  across  the  river  Brenta,  was  pre 
cisely  similar  in  principle  to  the  above.  This  bridge  also  was 
built  on  piles,  but  the  braces  and  straining-beams,  instead  of 
being  above  the  roadway  forming  part  of  the  balustrade,  were 
placed  below  and  framed  into  the  piles,  which  extended  up  to 
the  level  of  the  roadway.  This  bridge  was  surmounted  by  a 
roof  supported  by  Doric  columns,  connected  below  by  a  light 
handrail. 


BRIDGE  OVER  THE  TORRENT  AT  CISMORE. 

BY  PALLADIO.     Span  108  feet. 

FIG.  80. 


This  bridge  must  have  been  a  good  one  for  small  spans. 
The  arrangement  is  such  that  the  pressures  are  transmitted  to 
the  abutments  with  very  little  tendency  to  produce  a  change 
of  figure.  The  rise  at  the  point  A.,  which  would  be  produced 
by  the  action  of  a  weight  at  B,  is  counteracted  by  the  resist 
ance  of  the  tie  A  0. 

One  of  the  most  remarkable  designs  of  Palladio  consisted  of 
two  parallel  or  concentric  arches  connected  by  diagonal  braces. 


ILLUSTRATIONS   OF   PARTICULAR    MODES, 


As  it  appears  to  have  been  the  first  idea  of  constructing  a 
system  of  framed  voussoirs  similar  to  the  arch  stones  of  a 
bridge,  a  principle  that  has  been  adopted  to  a  considerable  ex 
tent  for  iron  bridges,  and  is  recommended  by  Tredgold  for 
structures  in  wood. 

In  Tredgold's  Carpentry,  will  be  found  a  plan  for  a  bridge 
of  400  feet  span  on  this  principle. 

The  objection  to  this  mode  of  construction  has  been  already 
stated.  The  only  points  that  bear  any  considerable  portion  of 
the  strain  are  B  and  Z>,  hence  the  timber  at  A  and  0  adds 
unnecessarily  to  the  weight.  Such  an  arch  would  undoubt 
edly  be  very  stiff,  and  would  oppose  great  resistance  to  change 
of  figure,  but  an  arch  is  not  the  only  element  required  in  the 
construction  of  a  roadway ;  it  is  merely  a  support  from  which 
vertical  pieces  must  extend  either  as  posts  to  support  a  road 
way  above,  or  as  ties  to  sustain  one  suspended  beneath.  In 
either  case  it  is  obviously  more  simple,  more  economical,  more 
elegant,  and  more  scientific,  to  make  a  single  arch  with  the 
maximum  rise,  and  secure  stiffness  by  counter-bracing  between 
the  uprights,  a  method  which  has  the  additional  advantage  of 
stiffening  the  uprights  themselves. 


FIG.  82. 


Another  design  from  Palladio  is  represented  above.  This 
truss  would  sustain  a  uniform  load,  but  would  not  suit  for  a 
viaduct  without  counter-bracinjr. 


144  BRIDGE   CONSTRUCTION. 

BRIDGE  ACROSS  THE  PORTSMOUTH  RIVER. 

Span  250  feet. 

Fia.  83. 


This  plan  is  giien  or,  Plate  16  of  Tredgold's  Carpentry,  ag 
a  specimen  of  an  American  bridge.  It  is  composed  of  three 
concentric  arcs,  connected  by  radial  pieces  without  either  braces 
or  counter-braces. 

Were  the  problem  given  us  to  arrange  a  given  quantity  of 
timber  in  the  most  unskilful  manner  possible,  it  would  be  diffi^ 
cult  to  select  a  plan  which  would  much  better  fulfil  the  required 
conditions.  By  separating  the  timbers  into  three  arches,  and 
placing  them  at  a  distance  apart,  the  whole  of  the  strain,  or  by 
far  the  greater  part,  is  thrown  upon  the  points  A  and  (7,  and 
only  one-third  of  the  material  is  so  disposed  as  to  resist  it. 
Again,  the  stiffness  of  such  a  system  would  be  little  more  than 
one-third  that  of  a  single  arch  containing  the  same  material, 
for  the  stiffness  being  as  the  square  of  the  depth  in  a  beam 
whose  depth  is  3,  it  will  be  represented  by  9,  and  in  a  beam 
whose  depth  is  1,  it  will  be  1.  Hence  3  beams  of  the  depth  1 
will  only  give  one-third  the  stiffness  of  a  single  beam  whose 
depth  was  equal  to  the  sum  of  the  three. 

Colonel  Douglass,  who  gives  a  description  of  this  work,  ob 
serves  that  the  arch  is  extremely  flexible.  This  result  would 
necessarily  follow  from  the  absence  of  counter-bracing. 

The  quantity  of  timber  must  have  been  very  great  to 
enable  it  to  stand  at  all,  if  heavy  variable  loads  were  drawn 
over  it. 


ILLUSTRATIONS   OF   PARTICULAR   MODES.  145 


TIMBER  BRIDGE  OVER  THE  RIVER  DON,  AT  DYCE  IN 
ABERDEENSHIRE. 

FIG.  84. 


We  were  much  surprised  upon  turning  to  the  article  Bridge, 
in  the  Edinburgh  Encyclopedia,  to  find  almost  the  identical 
plan  of  construction  which  our  theory  had  led  us  to  recom 
mend  as  best  adapted  to  bridges  of  large  span. 

This  structure  was  erected  by  Mr.  James  Burn  of  Hadding- 
ton,  near  Aberdeen,  in  the  year  1803.  The  description  does 
not  inform  us  in  reference  to  any  of  the  details  of  construction, 
and  we  cannot  tell  whether  the  architect  wedged  the  counter- 
braces  to  increase  the  stiffness  of  the  truss.  The  plan  of  using 
Avedged  counter-braces  appears  to  have  been  but  recently  in 
troduced,  and  forms  a  new  and  important  era  in  bridge  con 
struction;  even  yet,  many  practical  builders  do  not  seem  to 
understand  their  utility. 

SCHAFFHAUSEN  BRIDGE. 

FIG.  85. 


This  celebrated  structure  was  built  by  Ulric  Grubenmann, 
and  consisted  of  two  spans,  one  of  172  feet,  the  other  of  193. 
It  was  supported  in  the  interval  by  a  stone  pier,  which  had  re 
mained  when  a  former  bridge  had  been  swept  away.  With 
many  excellencies  this  bridge  had  also  serious  defects,  and  it 
is  certain  that  a  much  smaller  quantity  of  timber  judiciously 
arranged  would  have  given  far  greater  strength.  Still  the 
principle  is  an  admirable  one,  and  originating  as  it  did  with  an 
10 


BRIDGE   CONSTRUCTION. 


uneducated  village  carpenter,  certainly  displays  no  ordinary  ca 
pacity.  The  supports  consist  entirely  of  systems  of  arch-braces, 
but  the  details  were  too  complicated,  and  the  execution  evinced 
considerable  timidity.  It  would  seem,  from  an  inspection  of 
the  plan,  that  the  design  had  been  conceived  of  spanning  the 
whole  interval  at  once,  as  there  is  a  system  of  arch-braces  ex 
tending  from  the  abutments  towards  the  centre  ;  but  apprehen 
sive  that  such  a  long  interval  would  cause  the  bridge  to  fail, 
two  other  systems  of  arch-braces  were  introduced,  extending 
from  the  extremities  towards  the  centre  of  each  span. 

It  would  have  been  better,  either  to  have  spanned  the  wnole 
interval  by  one  magnificent  truss  of  365  feet,  which  could  have 
been  constructed  on  the  arch-brace  principle,  or  else  have  em 
ployed  two  separate  trusses,  one  for  each  interval. 

A  glance  at  the  figure,  which  exhibits  merely  the  general 
principle  without  attempting  to  represent  the  complicated  de 
tails,  will  show  that  it  was  destitute  of  counter-bracing  ;  and 
Mr.  Cox,  a  traveller  in  Switzerland,  states  that  "  a  man  of  the 
slightest  weight  felt  it  almost  tremble  under  him,  yet  wagons 
heavily  laden  passed  over  it  without  danger." 

Upon  the  principle  of  the  Schaffhausen  bridge  are  the  via 
ducts  of  the  Baltimore  and  Ohio  Railroad,  designed  we  believe 
by  B.  H.  Latrobe,  Esq.,  Chief  Engineer.  The  arch-brace  sys 
tem  is  here  combined  with  diagonal  ties  of  iron,  by  which  it 
is  effectually  counter-braced.  The  sizes  of  the  braces  are 
calculated  from  an  exact  estimate  of  the  weights  they  are  re 
quired  to  sustain,  and  the  whole  arrangement  and  proportion- 
ment  evince  a  thorough  acquaintance  with  the  subject,  and 
render  the  plan  admirably  adapted  to  span  any  interval,  or 
sustain  either  a  uniform  or  variable  load. 


LONG'S  BRIDGE. 

FIG.  86. 


ILLUSTRATIONS   OF   PARTICULAR   MODES.  147 

The  main  support  of  this  bridge  consists  of  a  system  of 
braces  and  ties,  and  in  large  spans  arch-braces  are  added ; 
keyed  counter-braces  are  also  used,  and  the  details  are  very 
well  arranged. 

These  bridges  have  been  extensively  used  on  many  of  the 
most  important  railroads  of  the  United  States.  In  the  city  of 
Baltimore  they  are  employed  at  the  crossings  of  most  of  the 
streets  that  intersect  the  direction  of  Jones'  Falls. 


LATTICE  BRIDGES. 


No  plan  of  bridge  construction  has  met  with  more  general 
favor  amongst  engineers  and  builders  than  the  lattice.  Its 
great  simplicity,  the  ease  with  which  it  can  be  framed,  and 
chiefly  its  economy,  have  secured  its  introduction  for  viaducts 
of  almost  every  class.  Of  late  years,  however,  the  frequent 
failures  of  these  bridges  in  consequence  of  heavy  transporta 
tion,  have  produced  a  revolution  of  sentiment  hostile  to  the 
plan,  and  instead  of  examining  into  the  causes  of  failure  and 
providing  a  remedy  for  the  defects  which  occasioned  it,  other 
modes  of  construction  have  been  adopted  at  an  expense  some 
times  double  that  of  an  efficient  lattice  structure. 

On  ordinary  roads,  and  on  railways  not  subjected  to  very 
heavy  transportation,  this  plan  of  superstructure,  when  well 
constructed,  has  been  found  to  possess  almost  every  desidera 
tum.  Nevertheless,  experience  has  fully  proved  that  unless 
strengthened  by  arch-braces  or  arches,  the  capacity  of  the 
structure  is  limited  to  light  loads,  and  spans  of  small  extent. 
The  public  works  of  Pennsylvania  furnish  abundant  proof  of 
the  truth  of -this  assertion;  and  several  railways  might  be 
enumerated,  on  which  the  lattice  bridges  have  from  necessity 
been  strengthened  by  props  from  the  ground,  by  arches,  or 
arch-braces  added  when  the  insufficiency  of  the  structure  was 
found  to  require  it. 

ri48) 


LATTICE   BRIDGES. 


149 


FIG.  87. 


\x\ 


xxxxxxxxx> 


The  lattice  truss  in  its  most  simple  form  consists  of  two 
sets  of  chords,  A  B  and  Q  D,  connected  by  diagonal  ties  and 
braces. 

The  chords  are  formed  of  plank  3  inches  X  12  inches, 
lapped  so  as  to  break  joint  both  above  and  below.  The  braces 
and  ties  are  also  of  3  inch  plank,  placed  between  the  chords 
and  pinned  with  wooden  pins  at  all  the  intersections.  From 
this  description  it  will  be  perceived  that  the  truss  possesses  the 
merit  of  simplicity  in  the  highest  degree. 

The  most  ordinary  carpenter  who  is  able  to  bore  a  hole 
with  an  augur  is  capable  of  constructing  it,  and  the  timbers 
being  all  of  the  same  size  are  delivered  from  the  mill  in  the 
state  in  which  they  are  put  together ;  hence  no  preparatory 
labor  is  required,  no  carefully  fitted  joints,  no  bolts,  straps,  or 
ties  of  iron,  and  consequently  it  is  fair  to  presume  that  the  lat 
tice  principle  is  the  cheapest  upon  which  a  truss-bridge  can 
be  constructed.  If,  therefore,  its  defects  can  be  removed,  we 
see  no  reason  why  this  mode  of  construction  should  not  take 
precedence  of  all  others  for  ordinary  purposes,  where  economy 
in  first  cost  is  an  object  of  importance. 

The  elements  of  the  lattice  truss  consist  of  horizontal 
chords,  and  inclined  ties  and  braces,  as  represented  in  the 


figure. 


FIG.  88. 


In  the  case  of  flexure,  the  pieces  in  the  direction  a  b  suf 
fer  compression,  and  therefore  act  as  braces,  and  those  in  the 
direction  c  d  are  extended  and  become  ties.  The  lattice 


150  BRIDGE   CONSTRUCTION. 

truss  therefore  possesses  this  peculiarity,  that  the  ties  are  all 
in  an  inclined  position,  instead  of  being  perpendicular  to  the 
chords,  as  in  other  modes  of  construction. 

That  this  inclined  position  of  the  ties  is  injurious,  we  are 
not  prepared  to  prove ;  although  several  considerations  lead 
us  to  suppose  that  it  is  less  efficient  than  when  the  ties  are 
perpendicular.  But  this  point  is  comparatively  unimportant, 
as  it  is  for  very  different  reasons  that  we  propose  a  change  in 
the  mode  of  construction. 

One  of  the  first  defects  apparent  in  old  lattice  bridges  is 
the  warped  condition  of  the  side  trusses.  The  cause  which 
produces  this  effect  cannot  perhaps  be  more  simply  explained, 
than  by  comparing  them  to  thin  and  deep  boards,  placed 
edgeways  on  two  supports,  and  loaded  with  a  heavy  weight. 
So  long  as  a  proper  lateral  support  is  furnished,  the  strength 
may  be  found  sufficient;  but  when  the  lateral  supports  are 
removed,  the  board  twists  and  falls.  A  lattice  truss  is  com 
posed  of  thin  plank,  and  its  construction  is  in  every  respect 
such  as  to  render  this  illustration  appropriate. 

A  second  defect  may  be  found  in  the  short  ties  and  braces 
at  the  extremities,  which,  furnishing  but  an  insecure  support, 
render  these  points,  which  require  the  greatest  strength,  weaker 
than  any  others;  this  defect  is  generally  removed  by  extend 
ing  the  truss  over  the  edge  of  the  abutment,  a  distance  about 
equal  to  its  height,  thus  providing  a  remedy  at  the  expense  of 
economy  by  the  introduction  of  from  15  to  30  feet  of  additional 
truss. 

Other  defects  can  be  mentioned,  which  are  not,  however, 
peculiar  to  lattice  bridges.  The  ties  and  braces  are  of  the 
same  size  throughout,  and  consequently  no  stronger  at  the 
point  of  greatest  strain  than  where  the  strain  is  least.  The 
same  remark  applies  also  to  the  chords.  Some  of  these  evils 
can  be  remedied  by  slight  additions.  By  bolting  arches  or 
arch-braces  to  the  truss,  the  weak  points  both  of  the  chords 
and  braces  can  be  effectually  relieved.  But  it  would  be  still 
better  to  depend  for  the  power  of  resisting  all  the  weight  upon 
an  arch-brace  system,  using  a  light  lattice  truss  only  as  a 
sounter-brace.  This  would  be  a  great  improvement ;  but  one 


LATTICE    BRIDGES.  151 

defect  would  still  remain  ;  there  would  not  be  sufficient  security 
against  warping,  although  much  more  than  in  the  ordinary 
method  of  construction. 

FIG.  89. 


The  double  lattice,  as  it  is  called,  consists  of  three  sets  of 
chords,  above  and  below,  as  represented  in  the  cross-section, 
between  which  two  sets  of  ties  and  braces  are  introduced.  In 
comparing  this  truss  with  the  single  lattice,  it  is  evident  that 
it  must  possess  greater  power  to  resist  warping,  for  the  timbers 
n  n  being  separated  by  an  interval,  will  act  on  the  principle 
of  a  hollow  cylinder,  which  is  much  stiffer  with  a  given  quan 
tity  of  material  than  a  solid  one.  This  however  is  its  only 
advantage,  in  other  respects  we  think  it  one  of  the  worst  that 
could  be  adopted.  Whilst  the  weight  of  timber  from  the  ties 
and  braces  has  been  doubled,  the  cross-section  of  the  chords 
has  been  only  increased  one-half.  A  great  load  of  unneces 
sary  timber  is  placed  in  the  centre,  where  any  weight  acts  with 
the  greatest  leverage,  and  produces  the  greatest  strain.  It  is 
probable  that  this  truss,  as  usually  constructed,  possesses  less 
absolute  strength  with  a  given  quantity  of  material  than  any 
other  in  common  use. 

The  greatest  improvement  that  could  be  made  to  this  truss,, 
would  be  to  introduce  two  arch-braces  and  a  straining-beam, 
and  the  opening  between  the  trusses  n  n  would  be  admirably 
adapted  for  the  reception  of  such  a  system. 

In  lattice  bridges  a  second  set  of  chords  is  sometimes,  per- 
haps  it  may  be  said  generally,  placed  between  the  first,  cross 
ing  the  second  intersections ;  but  as  these  chords  are  nearer 
the  neutral  axis,  they  of  course  act  less  efficiently  than  those 
which  are  at  the  top  and  bottom. 


IMPROVE])  LATTICE. 


THE  following  plan  of  a  bridge  truss  was  designed  by  the 
author  in  the  year  1840.  With  even  greater  simplicity  and 
economy  than  the  ordinary  lattice,  it  appears  to  be  entirely  free 
from  its  defects ;  and  possessing  many  of  the  essential  requisites 
of  a  good  bridge,  with  a  capability  of  extension  to  spans  of  con 
siderable  length,  it  seems  to  be  unusually  well  adapted  to  the 
wants  of  a  community  with  whom  economy  is  an  object. 

A  well  arranged  and  proportioned  structure  should  possess 
the  following  requisites : 

1.  The  cross-section  of  the  chords   should  be  greatest  at 
centre,  and  least  at  the  ends. 

2.  The  resisting    area  of   the  ties    and   braces   should   be 
greatest  at  the  abutments. 

3.  A  system  either  of  counter-braces  or  of  diagonal  ties  must 
be  introduced,  to  secure  the  structure  against  the  effects  of 
variable  loads. 

4.  The  timbers  of  the  side  trusses  should  be  Ox  such  a  size, 
or  arranged  in  such  a  manner,  as  to  guard  against  all  liability 
to  warp. 

5.  It  is  desirable,  although  not  always  necessary  or  practi 
cable,  that  the  pressures  should  be  divided  amongst  several 
timbers,  so  that  any  defective  piece  can  be  readily  removed 
and  its  place  supplied  by  another,  without  rendering  it  neces 
sary  to  support  the  bridge  during  the  progress  of  the  repair. 

(152) 


IMPROVED    LATTICE. 

FIG.  90. 


153 


In  the  improved  lattice  the  first  two  requisites  are  attained 
by  a  system  of  arch-braces  and  straining-beams,  which  is  the 
simplest  method  of  relieving  both  the  chords  in  the  centre,  and 
the  braces  and  ties  at  the  abutments.  Arches  are  preferable, 
but  rather  more  expensive. 

The  arrangement  of  the  intermediate  timbers  is  similar  to 
that  of  the  common  lattice,  and  the  manner  of  forming  the 
connections  by  wooden  pins  is  the  same ;  but  the  ties  instead 
of  being  inclined  are  vertical,  a  position  which  is  more  natural, 
more  efficient,  and  requires  less  material. 

The  braces  instead  of  being  single  are  reduced  in  size  and 
placed  in  pairs,  one  on  each  side  of  the  tie,  which  accordingly 
passes  between  them,  and  is  pinned  at  every  intersection. 

This  arrangement  secures  the  third,  fourth,  and  fifth  requi 
sites.  The  inclined  pieces,  from  the  manner  of  their  connec 
tion,  are  equally  capable  of  acting  as  braces  or  ties,  and  there 
fore  the  truss  is  counter-braced  by  a  system  of  diagonal  ties, 
without  the  necessity  of  introducing  timbers  expressly  for  this 
purpose,  as  in  most  other  plans. 

The  braces  being  in  pairs,  with  the  ties  passing  between, 
as  in  the  figure,  will  possess  the  stiffness  of  a  hollow  cylinder, 

FIG.  91. 


In  this  respect  it  possesses  the  only  good  quality  of  the 
double  lattice,  but  in  a  higher  degree,  for  there  are  here  inter 
mediate  points  a  and  5,  formed  by  the  passage  of  ties  to  which 
the  braces  are  pinned,  and  which  add  greatly  to  the  stiffness. 


154  BRIDGE   CONSTRUCTION. 

In  the  ordinary  lattice  the  braces  and  ties  being  8  inches  thick, 
if  they  were  placed  upon  each  other  in  the  same  direction, 
and  pinned  at  short  intervals,  the  stiffness  would  be  nearly  in 
proportion  to  the  square  of  6 ;  but  as  they  cannot  be  so  arranged, 
and  in  fact  cross  each  other  nearly  at  right  angles,  the  flexure 
of  one  system  is  not  affected  by  contact  with  the  other,  and  the 
lateral  stiffness  would  only  be  in  the  proportion  to  the  square 
of  3.  In  the  improved  plan  the  braces  are  made  2  inches  by 
10,  the  amount  of  timber  is  the  same  as  in  the  common  lattice, 
but  the  stiffness  would  be  nearly  in  the  proportion  of  the  square 
of  7,  or  five  times  that  of  the  common  lattice. 

It  is  evident,  too,  that  upon  the  removal  of  any  tie  or  brace 
the  weight  Avould  be  sufficiently  sustained  by  the  adjacent 
ones,  and  repairs  could  therefore  be  made  -without  difficulty, 
an  advantage  which  is  not  peculiar  to  this  plan,  but  is  pos 
sessed  also  by  several  others. 

In  addition  to  this  it  may  be  observed,  that  the  truss  does 
not  require  to  be  extended  back  any  considerable  distance 
from  the  face  of  the  abutment ;  there  are  no  short  ties  as  in  the 
common  lattice. 

The  mode  of  construction  which  has  been  designated  as 
the  improved  lattice,  admits  of  extension  to  any  span  to  which 
an  arch-braced  system  is  applicable,  but  is  exceeded  in  the 
length  to  which  it  might  be  extended  by  the  simple  counter- 
braced  arch.  In  very  large  spans,  whatever  be  the  general 
arrangement  of  the  timbers  of  the  truss,  the  whole  dependence 
for  the  support  of  the  structure  and  its  load  should  be  placed 
upon  the  arch-braces  and  the  straining-beams  which  join  the 
extremities.  This  system  may  be  connected  with  a  truss  on  the 
principle  of  the  improved  lattice,  by  which  it  will  be  effect 
ually  counter-braced  and  the  parts  properly  connected ;  such  an 
arrangement  is  represented  in  the  accompanying  figure,  and 
could  be  employed  for  long  spans. 

FIG.  92. 


IMPROVED   LATTICE. 


155 


COLUMBIA  BRIDGE. 

The  bridge  across  the  Susquehanna  river,  at  Columbia, 
consists  of  a  series  of  spans  of  about  200  feet  each,  the  whole 
length  of  the  bridge  being  about  1J  miles.  This  structure 
consists  of  a  truss  composed  of  braces  and  ties,  strengthened 
by  the  addition  of  an  arch,  and  although  the  bridge  is  straight 
the  upper  chords  are  not  continued  across  the  piers.  From 
the  absence  of  counter-bracing,  it  might  be  inferred  that  con 
siderable  vibration  would  be  produced  by  the  passage  of  a  load. 

This  is  in  fact  the  case ;  the  undulation  caused  by  a  pass 
ing  car  can  be  felt  at  a  distance  of  several  spans.  Many  of 
the  bridges  on  the  Philadelphia  and  Columbia  .Railroad  are  on 
the  same  principle.  They  are  very  light  structures,  but  the 
absence  of  counter-braces  is  an  objection.  The  following  fig 
ure  will  give  an  idea  of  the  plan. 


FIG.  93. 


The  old  bridge  across  the  Susquehanna  at  Harrisburg, 
one  half  of  which  remains,  is  similar  in  principle  to  that  at 
Columbia,  except  that  it  contains  heavy  counter-braces  of 
nearly  the  same  size  as  the  braces  themselves.  It  is  encum 
bered  with  unnecessary  timber,  but  in  other  respects  the 
arrangement  is  good. 

A  portion  of  this  bridge  was  recently  carried  away  by  a 
flood,  but  it  has  since  been  rebuilt.  The  railroad  bridge  across 
the  Susquehanna  at  the  same  place  is  on  the  double  lattice 
plan. 

An  arrangement  something  similar  in  appearance,  but  dif 
fering  altogether  in  principle  from  the  Columbia  bridge,  and 
which  would  possess  greater  stiffness,  consists  of  a  single  arch 
attached  to  a  counter-braced  truss.  No  doubt  can  be  enter 
tained  of  the  ability  of  the  arch  to  sustain  a  load  if  change  of 
figure  can  be  prevented ;  and  the  counter-braces  would  effect- 


156 


BRIDGE   CONSTRUCTION. 


ually  stiffen  it,  and  prevent  that  injurious  vibration  to  "which 
reference  has  been  made. 

FIG.  94. 


A  still  lighter  truss  could  be  formed  by  using  diagonal  ties 
instead  of  counter-braces.  The  vertical  pieces  would  then  be 
in  a  state  of  compression,  and  could  be  simply  notched  on  the 
chords  without  passing  through,  as  is  necessary  when  the  strain 
upon  them  is  one  of  extension.  This  arrangement  would  suit 
for  a  bridge  when  the  roadway  is  on  the  top  chord. 

Where  the  roadway  is  on  the  bottom  chord,  the  ties  should 
be  iron  rods  and  the  counter-braces  of  wood. 


Fia.  95. 


A  system  of  construction  applicable  to  spans  of  considerable 
extent,  consists  of  arch-braces  counter-braced  by  a  single  in 
verted  arch,  A  E  B.  The  arch  is  attached  by  iron  rods  pass 
ing  through  the  straining-beams  with  nuts  on  the  top.  The 
nuts  being  at  the  top  of  the  truss  would  be  at  all  times  acces 
sible,  the  strain  could  be  regulated  at  pleasure.  The  long 
braces  at  the  ends  would  require  intermediate  supports. 

Instead  of  the  inverted  arch,  an  ordinary  counter-braced 
truss  consisting  of  chords,  ties,  and  counter-braces,  without 
braces,  could  be  used. 

Trussed  girder  bridges  which  consist  of  two  or  more  lon 
gitudinal  timbers,  strengthened  by  iron  rods  passing  beneath 
them  and  adjustible  by  screws,  are  strong,  cheap,  and  when 
properly  constructed  and  proportioned,  are  very  efficient.  As 


IMPROVED    LATTICE.  157 

usually  constructed,  with  two  posts  dividing  the  span  into 
three  intervals,  they  are  without  diagonal  rods  or  braces  in  the 
middle  interval ;  this  is  a  defect  which  should  be  avoided. 
For  considerable  spans,  the  intervals  must  be  increased  in 
number,  and  the  figure  of  the  truss  becomes  a  polygon,  bounded 
by  a  straight  chord  on  the  upper  side,  and  by  one  or  more  iron 
rods,  forming  a  broken  line  on  the  lower  side. 

An  application  of  this  principle,  which  does  not  appear  to 
have  been  made,  but  which  would  be  useful  in  many  cases, 
consists  in  trussing  the  top  instead  of  the  bottom  chord. 
Trussed  girder  bridges  could  then  be  used  when  the  roadway 
passes  through  the  bridge,  as  well  as  when  it  passes  over  the 
top.  In  this  case,  the  top  chord  must  be  well  braced  laterally, 
and  the  ends  must  be  supported  by  strong  posts. 

One  of  the  simplest,  and,  for  an  iron  bridge,  one  of  the 
cheapest  modes  of  construction,  consists  in  using  a  single  arch, 
a  straight  top  chord,  and  vertical  posts,  or  columns  connecting 
the  chord  and  the  arch  without  panel  braces  or  ties  of  any 
kind,  and  without  a  Uwsr  chord.  The  arch  is  counter-braced 
by  iron  rods  extending  from  the  chords  over  each  post  to  the 
abutments  below  the  skew-backs,  where  they  are  securely 
anchored  into  irons  passing  through  the  masonry. 

Many  other  arrangements  and  combinations  might  be  given, 
but  as  the  object  of  the  author  in  the  first  part  has  been  to  es 
tablish  general  principles,  and  not  to  exhibit  details,  the  reader 
is  permitted  to  exercise  his  ingenuity  in  making  other  combina 
tions  rf  the  elements  of  bridge  trusses,  viz.,  chords,  ties,  braces, 
counter-braces,  arches,  and  arch-braces. 


PAET  II. 


PREFACE   TO   SJECOiND   PART. 


CONSIDERABLE  time  lias  elapsed  since  the  preparation  of  tbe 
former  part  of  this  work,  during  which  so  many  improve 
ments  have  been  introduced  into  the  practice  of  bridge 
construction,  that  a  further  extension  of  the  descriptions 
of  particular  plans  seems  to  be  necessaiy.  It  is  believed 
that  no  work  has  ever  been  published  containing  detailed 
calculations  of  the  strains  upon  all  the  timbers  which  con 
stitute  the  supporting  trusses  of  a  framed  bridge,  nor  has 
any  theory  been  advanced  which  furnishes  rules  by  which 
these  strains  can  be  estimated.  At  the  suggestion  of 
several  professional  friends,  who  concurred  in  the  opinion 
that  such  an  addition  was  a  desideratum,  the  writer  was 
induced  to  prepare  this  second  part,  containing  details  of 
most  of  the  arrangements  that  are  exhibited  in  wooden 
structures,  furnishing  illustrations  of  all,  or  nearly  all,  the 
11  (161) 


162  PREFACE   TO   SECOND   PART. 

different  modes  or  forms  of  calculation  that  can  be  required 
in  estimating  the   strains  upon  bridges,  including  combi 
nations  of  several  systems. 

There  may  be,  and  no  doubt  will  be,  differences  of 
opinion  in  reference  to  the  proper  method  of  estimating 
the  strains  where  several  systems  are  united,  and  it  is 
admitted  that  the  solution  of  the  problem  must  be  based 
upon  hypotheses  which  may  not  always  exhibit  the  true 
practical  conditions  of  the  case.  By  supposing  all  the 
iointfl  to  bear  eaually,  it  may  be  assumed  that  each  system 
contributes  to  sustain  the  load  in  proportion  to  its  powers 
of  resistance.  This  never  can  be  practically  true ;  joints 
cannot  be  made  perfect;  and  consequently,  a  calculation 
made  upon  this  hypothesis  must  be  to  some  extent  erro 
neous.  Notwithstanding  this  objection  there  is  no  better 
way,  and  when  it  is  considered  that  the  material  is  gener 
ally  elastic,  and  that  those  joints  which  are  most  tightly 
compressed  at  first  will  yield,  and  bring  others  into  more 
intimate  contact,  the  error  will,  after  all,  not  be  of  much 
practical  importance,  and  the  results  will  furnish  a  safe 
guide  in  proportioning  the  parts  of  structures. 

A  proper  attention  to  his  professional  duties  has  not 
allowed  the  author  time  for  careful  revision  of  the  calcula 
tions;  it  is  believed,  however,  that  no  errors  exist  that 
affect  any  of  the  general  principles,  and  if  mistakes  should 
be  found  in  any  of  the  numerical  results,  it  must  be  re 
membered  that  these  calculations  are  given  merely  aa 
illustrations,  and  great  accuracy  has  not  been  considered 
necessary. 


PREFACE  TO  SECOND  PART.  163 

The  construction  of  the  Pennsylvania  Railroad  requiring 
a  large  amount  of  bridge  superstructure,  has  afforded  an 
opportunity  for  the  introduction  of  various  plans,  some  of 
which  are  new,  and  all  have  been  so  fully  tested  by  expe 
rience  or  based  upon  such  well-tried  principles,  that  they 
can  confidently  be  relied  upon.  The  description  of  these 
plans  with  some  of  those  in  use  upon  other  roads,  will 
furnish  a  very  satisfactory  exposition  of  the  present  state 
of  the  science. 

It  will  be  observed  that  no  particular  mode  of  con 
struction  is  advocated  in  this  work,  but  efforts  are  made  to 
illustrate  and  establish  the  general  principles  that  must 
govern  the  engineer  in  every  case.  The  following  are  the 
most  important : 

1.  In  a  straight  bridge  uniformly  loaded,  and  without 
arches  or  arch-braces,  the  strain  upon  the  ties  and  braces  at 
the  middle  point  of  the  bridge  is  almost  nothing. 

2.  The  strain  at  each  end  upon  the  same  timbers  is 
equal  to  half  the  whole  weight  of  the  bridge  and  its  load. 

3.  The  strain  at  intermediate  points  is  proportional  to 
the  distance  from  the  middle  of  the  span, 

4.  "Wliere  the  bridge  is  subjected  to  the  action  of  varia 
ble  loads,  the  greatest  strain  on  the  ties  and  braces  at  the 
middle  of  the  span  is  equal  to  the  greatest  variable  load 
that  can  be  applied  in  the  interval  of  one  panel. 

5.  The  strain  upon  the  chords  is  greatest  in  the  mid 
dle,   and   at   this  point  is   dependent  entirely  upon  the 
vreight,  span,  and  depth  of  the  truss;  the  inclination  of  the 


164  PREFACE   TO   SECOND  PART. 

braces  lias  no  influence  upon  the  maximum  strain  upon  the 
chords. 

6.  "Where  there  is  only  one  span,  or  where  in  continuous 
spans  the  top  chords  are  not  connected  over  the  piers,  the 
strain  on  the  end  of  the  chord  is  nothing,  but  at  the  end  of 
the  first  brace  it  is  equal  to  one-half  the  weight  on  each 
truss  multiplied  by  the  cosine  of  the  inclination  of  the 
brace  from  a  horizontal  line. 

7.  "Where  the  weight  of  the  bridge  is  constant  and  uni 
form,  counter-braces  are  unnecessary;   but  for  viaducts, 
where  the  load  is  variable  and  unequally  distributed,  they 
are  indispensable. 

These  principles  indicate  the  most  important  conditions 
which  every  well-proportioned  structure  should  fulfil ;  any 
structure  built  in  accordance  with  them,  and  having  its 
parts  properly  proportioned  to  the  strains,  must  give 
satisfactory  results. 

The  difficulty  of  procuring  perfectly  accurate  drawings 
and  details  is  so  great,  and  the  time  which  the  author  has 
been  able  to  devote  to  this  work  so  small,  that  he  has  been 
compelled  in  some  instances  to  supply  unimportant  omis 
sions,  by  inserting  what  he  believed  to  be  necessary  or  to 
have  been  used  in  that  particular  case.  For  example,  the 
plans  forwarded  would  sometimes  be  defective  in  details  of 
lateral  bracings,  arrangement  of  flooring,  size  of  plank,  &c. ; 
instead  of  troubling  contributors  by  writing  for  further  ex 
planations,  these  deficiencies  have  been  supplied  as  above 
stated.  This  course  will  not  be  considered  objectionable, 


PREFACE   TO   SECOND   PART.  165 

when  it  is  remembered  that  the  object  of  this  pail  of  the 
work  is  chiefly  to  furnish  practical  illustrations  of  the  mode 
of  calculation. 

For  the  purpose  of  comparing  the  cost  of  different  struc 
tures,  an  estimate  has  been  made  in  each  case  for  a  single 
track  railroad  bridge,  16  feet  wide  between  trusses,  at  the 
prices  now  paid  for  work  gn  the  Pennsylvania  Railroad  ;  in 
preparing  an  estimate  from  them,  an  engineer  will  of 
course  vary  these  prices  to  suit  his  locality. 

In  making  the  calculations,  only  the  dead  weight  has 
been  considered ;  the  effect  of  the  momentum  of  a  passing 
load  depends  so  much  upon  contingencies,  irregularities 
of  surface,  &c.,  that  no  attempt  has  been  made  to  calculate 
it.  It  is  proper  to  add  from  25  to  50  per  cent,  in  railroad 
bridges  to  compensate  for  these  effects. 

The  deterioration  of  the  resisting  powers  of  timber 
caused  by  age  must  also  be  considered.  After  a  wooden 
bridge  has  been  in  use  for  some  years,  it  becomes  much 
weaker  than  when  erected.  The  allowance  made  for  the 
safe  limit  of  the  resisting  power  of  wood  is  1000  pounds 
per  square  inch  and  of  iron  10,000  pounds,  but  it  is 
probable  that  800  pounds  per  square  inch  for  wood,  and 
8000  pounds  per  square  inch  for  the  tensile  resistance  of 
large  rods  of  malleable  iron,  would  be  more  nearly  the  truo 
medium  between  economy  and  safety;  of  this,  however, 
every  engineer  must  judge  for  himself.  It  is  very  certain 
that  there  is  no  economy  in  risk, — an  excess  of  strength  is 
far  better  than  a  deficiency. 


166  PREFACE    TO   SECOND   PART. 

The  tables  of  data  given  "by  Tredgold  and  others,  in 
which  more  than  3000  pounds  per  square  inch  is  given  as 
the  strain  that  timber  will  bear,  is  liable  to  mislead  the 
inexperienced.  The  writer  has  observed  that  students,  in 
making  calculations  from  such  data,  often  arrive  at  con 
clusions  which  a  practical  man  would  consider  as  super 
latively  ridiculous.  Good  specimens  will  bear  for  a  short 
time  even  more  than  this,  but  great  allowances  must  be 
made  in  practice. 


PENNSYLVANIA  RAILROAD  VIADUCT,  ACROSS  THE 
SUSQUEHANNA  RIVER.     (Plate  1.) 


Description. 

THIS  structure  is  located  on  the  line  of  the  Pennsylvania  Rail 
road  five  and  a  half  miles  north  of  Harrisburg,  and  crosses  the 
river  Susquehanna  at  an  angle  of  68J°  with  the  general  direc 
tion  of  the  stream.  It  is  supported  on  22  piers  and  2  abutments. 
The  piers  are  founded  upon  cribs,  which  is  the  usual  method 
of  building  in  the  Susquehanna,  and  experience  has  proved  it 
to  be  perfectly  secure,  no  instance  having  ever  occurred,  so  far 
as  the  information  of  the  writer  extends,  of  the  failure  of  a  crib 
foundation  in  this  river.  The  cribs  of  the  Pennsylvania  Rail 
road  Viaduct  consist  generally  of  only  one  course  of  timber, 
12  X  12,  framed  sufficiently  wide  to  extend  2  feet  beyond  the 
line  of  the  regular  masonry,  and  sunk  sufficiently  low  to  be  at 
least  six  inches  below  the  surface  of  extreme  low  water.  The 
timbers  are  connected  by  cross-pieces  dovetailed  into  them. 
The  compartments  of  the  cribs  between  the  cross-pieces  are 
filled  with  large  and  small  stone,  laid  compactly  without  mor 
tar  to  a  level  with  the  upper  surface  of  the  crib ;  upon  this  is 
placed  the  foundation  course  of  the  masonry,  consisting  of  very 
large  stones  from  20  to  24  inches  high,  forming  a  regular 
course  upon  which  the  cement  masonry  is  commenced  with 
an  offset  of  6  inches. 

C167) 


168  BRIDGE   CONSTRUCTION. 

The  character  of  the  masonry  is  rock  range  work  laid  in 
hydraulic  cement.  Each  pier  contains  on  an  average  420 
perches  of  dressed  stone ;  the  width  of  the  piers  is  6  feet  on 
top  and  10  feet  at  the  springing  line  of  the  arches,  measured 
perpendicularly  to  the  general  direction  of  the  pier.  The 
foundation  is  protected  by  rip-rap  of  loose  stone,  which  becomes 
continually  more  solid  by  the  action  of  freshets,  the  effect  of 
which  is  to  deposit  sand  and  gravel  in  the  interstices.  Each 
pier  is  furnished  with  an  ice-breaker,  the  slope  of  which  is  45°. 
The  ice-breakers  are  built  with  the  same  kind  of  stone  work 
as  the  piers  tnemseives ;  heavy  oak  timbers  are  ancnorea  across 
the  face  of  each,  to  which  the  oak  facing  timbers  are  securely 
spiked. 

The  first  pier  was  covered  with  bars  of  cast-iron  placed 
longitudinally,  and  about  12  inches  apart,  the  spaces  between 
being  filled  with  concrete.  An  unexpected  period  of  cold 
weather,  immediately  after  the  concrete  was  laid,  caused  it  to 
freeze  before  setting,  and  a  freshet  at  the  same  time  washed 
out  a  portion  at  the  lower  end ;  so  that  the  result  was  not  as 
satisfactory  as  under  other  circumstances  it  would  have  been. 
This  mode  of  facing  an  ice-breaker  is  economical,  and  secure 
against  fire,  which  might  be  communicated  to  the  bridge  by 
coals  falling  upon  it,  blown  off  the  floor  by  the  force  of  wind. 
Ten  of  the  piers  are  covered  with  long  oak  timbers  10  X  10, 
laid  so  as  to  leave  openings  between  of  one  inch,  which,  when 
the  timber  has  become  completely  dry,  will  be  filled  with 
cement  as  security  against  fire.  The  ice-breakers  of  the  eleven 
remaining  piers  are  covered  with  bars  of  flat  iron  secured  as 
follows :  holes  were  punched  in  the  iron  bars  at  intervals  of 
2  J  feet,  sufficiently  large  to  receive  bolts  |  inch  diameter.  The 
bolts  were  6  inches  long,  split  for  half  their  length  to  receive  a 
wedge.  Holes  were  drilled  in  the  stones  at  proper  intervals 
to  receive  the  bolts.  These  holes  were  filled  with  ordinary 
mortar  of  sand  and  cement,  poured  in  before  driving  the  bolts. 
This  mode  appears  to  answer  perfectly ;  it  is  much  less  ex 
pensive  than  lead  and  more  convenient  of  application. 

The  foundations  of  the  22  piers  were  commenced  and 
carried  above  water  in  16  weeks ;  but  after  the  eighth  pier  was 


HAILROAD   VIADUCTS.  169 

founded,  operations  were  suspended  for  a  period  of  two  months 
not  included  in  this  statement.  With  the  exception  of  a  small 
portion  of  material  that  had  been  previously  delivered,  the 
whole  of  the  work  on  the  piers  and  abutments,  including  an 
arcade  of  three  full  centre  arches  of  25  feet  span,  constituting 
the  eastern  approach  of  the  viaduct,  was  completed  in  one 
season. 

The  whole  amount  of  stone  work  in  the  piers  and  abut 
ments,  including  three  abutments  of  a  single  span  bridge 
across  the  Pennsylvania  Canal  at  the  eastern  end  of  the  via 
duct,  and  the  cribs  of  the  foundations,  is  17,000  perches,  of 
which  nearly  13,000  perches  are  built  of  rock  range  work* 
The  final  estimate  for  the  masonry  was  $96,355  84. 

Each  pier  cost  about  §3,500. 


Superstructure. 

The  superstructure  of  the  Pennsylvania  Railroad  Viaduct 
is  built  upon  Howe's  plan,  with  the  addition  of  substantial 
wooden  arches.     The  spans  are  160  feet  from  centre  to  centro 
of  piers.     The  following  table  gives  the  principal  dimensions. 
Span  from  skew-back  to  skew-back  149  ft.     3  in. 

Versed  sine  of  lower  arch  20  "    10  " 

.  Under  side  of  skew-back  below  bottom  chord      6  " 
No.  of  panels,  16. 

Pier  panel  3  "    10  " 

Distance  between  wall  plates  on  piers  4  "      8  " 

Length  of  panels  9  "      9  " 

Width  in  clear  between  arches  13  "    11  " 

"  "  "        chords  15  "      5  " 

"      from  out  to  out  of  chords  19  " 

"       "       "         "         arches  20  "      6  « 

Angle  of  pier  and  chord  68J° 

Thickness  of  pier  at  right  angles  to  skew-back  10  ft. 
The   arches   are   in  3  segments,  the  dimensions   being  at 
centre  11  +  7  +  11  deep  by  9  inches  wide,  at  skew-back  11 
4-11  +  11  deep  by  9  inches  wide. 


L70  BRIDGE   CONSTRUCTION. 

Hypothenuse  of  skew-back  33     in 

Perpendicular  "  29-2  " 

Base  "  15-3  " 

Height  of  truss  from  out  to  out  of  chords  18  ft. 

o 

After  the  14th  span  had  been  raised,  a  violent  tornado  oc 
curred,  March  27th,  1849,  which  carried  off  six  spans.  These 
spans  were  in  an  unfinished  condition.  The  contractor  was 
engaged  at  the  time  in  putting  in  the  arches,  and  as  the  diag 
onal  braces  could  not  be  permanently  introduced  until  after  the 
arches  were  in  place,  he  had  omitted  them,  except  over  the 
piers  and  in  the  middle  of  the  spans.  The  direction  of  the 
storm  was  nearly  at  right  angles  to  the  bridge.  The  failure 
commenced  at  the  extreme  end  which  was  supported  on  tres 
tles.  The  bridge  gave  way  by  falling  together  in  the  direction 
of  the  diagonal.  The  only  arrangement  that  could  have 
secured  the  bridge  in  so  violent  a  tornado,  would  have  been  a 
complete  system  of  diagonal  bracing,  but  the  accident  occurred 
before  these  could  be  introduced,  in  consequence  of  the  un 
finished  condition  of  the  arches. 

Experiments  were  made  by  the  writer  to  ascertain  whether 
it  would  have  been  possible  for  the  wind  to  carry  away  the 
bridge  by  sliding  along  the  top  of  the  pier  or  wall  plate,  but 
the  least  friction,  in  an  average  of  15  or  20  experiments,  was 
T72  of  the  pressure,  which  was  sufficient  to  produce  a  resistance 
4  times  as  great  as  the  force  of  the  wind  upon  the  exposed 
surface,  at  that  time,  estimating  the  force  of  wind  at  14  pounds 
per  square  foot.* 

*  It  is  desirable  that  further  experiments  should  be  made  to  ascertain  the 
force  of  the  wind  in  violent  storms.  It  is  probable  that  it  is  generally  under 
rated.  The  -writer  addressed  letters  to  gentlemen  who  had  been  engaged  in 
making  observations  with  the  anemometer.  The  most  satisfactory  answer 
was  given  by  Professor  Bache,  who  stated  that,  "  on  Saturday,  August  5th, 
1843,  at  8  o'clock  P.  M.,  a  tornado  passed  within  a  quarter  of  a  mile  of  the 
Observatory  (Girard  College),  and  the  force  of  wind  was  so  great  as  to  exceed 
the  range  of  the  spring,  and  to  break  the  wire  connecting  it  with  the  plate 
of  the  anemometer ;  the  force  required  exceeded  42  pounds  to  the  square 
foot,  which  was  the  range  of  possible  movement  of  the  registering  arm.  The 
next  greatest  force  of  wind  was  14  pounds  to  the  square  foot,  from  4  to  5 
o'clock,  A.M.,  on  the  17th  Feb.  1842.  From  0  to  5  hours,  A.  M.  on  the  same 


RAILROAD  VIADUCT. 


171 


Bills  of  Materials  for  one  span  Pennsylvania  Railroad 
Viaduct. 


WHITE 

OAK. 

4 

Wall  plates 

8 

X 

12 

21 

ft. 

long 

672 

4 

Bolsters 

8 

X 

10 

17* 

u 

u 

466 

12 

Braces 

7 

x 

7 

19 

(C 

u 

932 

Total  Oak 


2,070 


WHITE   PINE. 


34 

Chord  pieces 

5 

x 

12 

39   ft. 

long 

6630 

17 

«            a 

5 

x 

10 

39    « 

" 

2764 

81 

a             u 

10 

X 

10* 

39    " 

« 

2900 

64 

Main  braces 

6 

X 

7 

19    « 

a 

4262 

32 

Counter  " 

6 

X 

7 

19    " 

u 

2131 

64 

Lateral   " 

5 

X 

6 

18*  « 

a 

2963 

44 

Floor  beams 

7 

X 

14 

24    " 

(t 

8624 

30 

Diagonal  braces 

5 

X 

6 

23J  « 

a 

1770 

12 

Track  strings 

9 

x 

11 

29*  " 

" 

2920 

72 

Arch  pieces 

9 

X 

11 

27    " 

« 

16056 

60 

Purlines 

2* 

X 

4 

16    •* 

a 

804 

20 

u 

6 

x 

12 

16    " 

it 

1920 

3200 

feet  inch  boards 

10    « 

tt 

3200 

Total  no.  of  feet  B.  M.  59,014 

"      cubic  feet  4,918 

No.  cubic  feet  per  foot  lineal  30-7 

"Weight  of  timber  per  foot  lineal,  1105  pounds. 

day,  the  mean  was  12'6  pounds.  From  0  to  11  hours  A.  M.  the  mean 
was  11'4  pounds,  and  for  the  day  7*69  pounds." 

The  above  is  an  extract  from  the  letter  of  Professor  Bache.  The  ob 
server  at  the  instrument  at  the  time  of  the  tornado  was  Mr.  Lewis  L. 
Haupt  (brother  of  the  writer),  who  says  that  he  has  a  distinct  recollec 
tion  of  the  occurrence,  and  that  the  wire  of  the  anemometer  broke  by 
the  force  of  a  sudden  blast  of  wind,  at  the  instant  when  it  registered  30 
pounds. 

Further  information  on  this  subject  is  very  desirable. 


172  BRIDGE   CONSTRUCTION. 


Bill  of  Castings  for  one  span. 

30  Bottom  chord  angle  blocks,  each  90  Ibs.  =  2,700 

30  Top           "  "          "  86  "  =  2,580 

8  Half  angle  blocks  "  58  «  -  464 

26  Bottom  gibs,  4  holes  "  34J  «  =  897 

4          "        "     2     "  "  25  "  100 

34  Top         "     2     «  «  20  «  =  680 

68  Lateral  angle  blocks  "  13  "  =  884 

204  Combination  keys  «  9  "  =  1,836 

68  Lateral  bolt  washers  "  3  "  204 

236  |  inch  washers  "        -J  "  =  206 

10,551 

Bill  of  bolts  for  one  span,  exclusive  of  nuts  and  washers. 
236     |  inch  bolts,  each      3|  Ibs.  =    855  each    2  ft.  1  in.  long. 


34 

14 

<c 

a 

"   64f 

u 

=  2001  " 

19 

a 

G 

" 

" 

8 

n 

U 

u 

«   75 

u 

=  600  " 

19 

u 

C 

a 

u 

20 

li 

U 

a 

"  112 

a 

=  2241  « 

18 

« 

7 

a 

a 

16 

If 

U 

u 

«  127 

a 

=,2032  « 

18 

u 

7 

a 

" 

24 

If 

a 

u 

"  151 

a 

=  3624  « 

18 

u 

7 

" 

u 

11,353 

Suspension  bolts. 


8 

If  diam. 

each  32J 

Ibs. 

=  260 

6 

ft. 

6 

in, 

8 

if 

a 

"   50 

(( 

=  400 

10 

a 

1 

a 

8 

if 

u 

"   65 

(£ 

=  520 

13 

it 

1 

a 

8 

it 

a 

«   77 

a 

=  616 

15 

ii 

6 

" 

8 

if 

« 

«   85 

" 

=  680 

17 

u 

1 

a 

8 

if 

u 

«   90 

u 

=  720 

18 

a 

4 

ii 

u 

"   93 

a 

=  372 

18 

« 

6 

u 

Weight  of  arch  bolts      3568 

Total  weight  of  bolts  14,920  pounds, 


RAILROAD   VIADUCT.  173 

Weight  of  nuts  for  one  span. 

236  nuts  for  f  inch  bolts,  each  -J  pound  =  118 

68  "  1J-  "  "  1TV  "  =  75 

16  «  1J  "  "  1-4  "  =  23 

40  "  1J  «  "  3-1  "  =  124 

104  "  If  "  "  2-0  "  =  208 

32  "  If  "  "  4-3  "  =  138 

48  «  If  "  "  4-6  "  =  221 

Total  weight  of  nuts   907  Ibs. 

Estimate  of  cost  of  one  span. 

2070  feet  B.  M.  oak  scantling  @  $18  00  per  M.  $  37  26 

56944    "        "      white  pine       @     13  00     "    "  740  27 

10551  Ibs.  castings                       @           02J  263  77 

14920    "    bolts                          @          04  59680 

907    "    punched  nuts              @          09  81  63 

4000    "    square  feet  roofing     @           08  320  00 

Total  cost  of  materials  §2039  73 

Workmanship. 

160  lineal  feet  superstructure  @  $5  50  $880  00 

236  |  inch  bolts,  head  at  one  end  @  10  23  60 

34  1J        "        screws  at  each  end  @  35  11  90 

8  1J        «                     «  @  45  3  60 

52  If        "                     "  @  55  28  60 

20  1J        u                     "  @  65  13  00 

16  If        «                     «  @  75  12  00 

24  If        "                     «  @  85  2040 

4000  square  feet  metal  roofing  @  02  80  00 

painting,  &c.  100  00 

$1175  10 

Total  cost  of  materials  and  work,  $3214  83.  —  Cost  per 
foot  lineal,  §20  00. 


174  BRIDGE   CONSTRUCTION. 

Principles  of  calculation. 

Before  we  proceed  to  calculate  the  strains  upon  the  parts 
which  compose  this  truss,  it  is  necessary  to  state  distinctly  the 
principles  upon  which  such  calculation  must  be  made.  It  is 
evident  that  where  two  systems  are  connected  in  the  same 
truss,  each  capable  of  opposing  a  certain  resistance,  it  will  be 
very  difficult  so  to  proportion  the  weight  upon  each,  that  the 
load  will  be  in  proportion  to  the  strength  of  the  several  portions. 
If,  for  example,  a  truss  be  constructed,  and  the  false  works  re 
moved  before  the  introduction  of  the  arches,  if  the  latter  be 
bolted  to  the  posts,  the  weight  of  the  whole  structure  is  sus 
tained  by  the  truss  itself,  and  the  arches  will  not  bear  a  single 
pound,  unless  they  are  called  into  action  by  an  increased  de 
gree  of  settling  in  the  truss.  But  if  the  bottom  chord  of  the 
truss  is  connected  with  the  arches  by  means  of  suspension  rods 
with  adjusting  screws,  the  whole  truss  may  be  raised  upon  the 
arches,  and  in  this  case  the  latter  will  bear  the  whole  weight, 
and  the  former  none. 

Again ;  if  we  suppose  the  arches  to  be  connected  with  the 
truss  before  the  removal  of  the  false  works,  and  the  joints  be^ 
equally  perfect  in  both  systems,  there  is  a  prospect  of  a  more 
nearly  uniform  distribution  of  the  load ;  but  even  in  this  case, 
we  cannot  tell  what  portion  is  sustained  by  each  system,  be 
cause  this  will  depend  upon  their  relative  rigidity.  If,  for  ex 
ample,  one  of  the  systems  should  experience  double  the  deflec 
tion  of  the  other,  with  a  given  load,  the  less  flexible  would 
sustain  twice  as  much  as  the  other  when  combined,  provided 
they  are  so  nicely  adjusted  as  to  bear  equally  when  unloaded 
except  with  the  weight  of  the  structure. 

In  practice,  the  most  convenient  way  of  securing  an  equal 
bearing  appears  to  be,  to  remove  the  false  works  before  the 
arches  are  introduced.  After  the  arches  are  in  place,  examine 
the  level  of  the  roadway,  and  screw  the  nuts  of  the  suspension 
arch  rods  until  the  truss  begins  to  rise  very  slightly.  As  there 
is  necessarily  a  certain  degree  of  elasticity  in  the  truss,  it  will 
then  be  certain  that  both  systems  are  in  action. 

With   all   these   precautions,  there  are  still   difficulties   in 


RAILROAD  VIADUCT.  175 

estimating  the  exact  strain  upon  the  parts  of  a  bridge  which  is 
sustained  by  two  different  systems ;  for  there  may  be  unequal 
settlement,  and  the  adjustment,  however  accurately  made  in 
the  first  place,  may  not  long  continue.  It  can,  it  is  true,  be 
tested  at  any  time,  by  unscrewing  the  suspension  bolts  until 
the  truss  ceases  to  settle,  and  then  screwing  up  again  until  the 
truss  begins  to  rise ;  but  it  will  generally  happen  that  after  a 
bridge  has  been  a  long  time  in  operation,  the  two  systems  bear 
very  unequal  portions,  and  when  the  truss  itself  is  not  so  con 
structed  as  to  be  susceptible  of  adjustment,  the  arch  almost 
always  sustains  the  whole  weight  of  the  bridge,  and  its  load. 

These  and  many  other  considerations  have  led  the  writer  to 
the  conclusion  that  the  best  method  of  constructing  bridges  is 
to  place  the  entire  dependence  upon  the  arch,  using  the  truss 
merely  as  a  system  of  counter-bracing  and  a  support  to  the 
roadway. 

In  the  structure  now  under  consideration,  either  the  truss 
without  the  arches,  or  the  arches  without  the  truss,  would  be 
sufficient  to  bear  the  load. 

The  calculation  of  the  strength  will  be  made  on  three 
hypotheses : 

1.  That  the  arch  sustains  the  whole  weight. 

2.  That  the  truss  sustains  the  whole  weight. 

3.  That  the  arch  and  truss  together  form  one  system. 

1st.  Calculation  of  the  strength  of  the  bridge  on  the  supposition 
that  the  arch  sustains  the  whole  weight. 

The  data  required  in  this  case  are, 

Distance  of  centre  of  gravity  of  half  truss 
from  abutment  37J  feet. 

Distance  from  centre  of  pressure  of  arch 
at  skew-back,  to  centre  of  pressure  at  crown  20  g  " 

Cross  section  of  arches  in  middle  of  span         1044   sq.  in. 

Cross  section  of  arches  at  skew-back  1188      " 

Weight  of  half-span  1281  pounds  per  foot,  102,500  pounds. 

Load  on          "          2000  "  160,000       « 

Total  load  262,500  Ibs. 


176  BRIDGE   CONSTRUCTION. 

262,500  x  37-5 
9Q-88  x  1044    :  ~  PreS8ure  on  arches  per  square  inch  at  cen 

tre  =  453  Ibs.  or  about  |  the  crushing  weight. 

The  pressure  in  the  direction  of  the  tangent  of  the  circle  at 
the  skew-back  bears  to  the  pressure  in  the  middle,  the  propor 
tion  of  the  hypothenuse  to  the  perpendicular  of  the  skew-back  ; 

453  x  1044  x  33 
it  will  therefore  be  —          x  29-9  —  =  ^s''  or  almost 


precisely  the  same  pressure  per  square  inch,  as  at  the  middle 
of  the  span.  It  appears,  therefore,  that  the  proportion  between 
the  cross-section  of  the  arches  at  the  ends  and  at  the  centre  is 
very  exact,  and  that  the  arches  alone  are  sufficiently  strong  to 
bear,  with  perfect  safety,  3  or  4  times  the  greatest  load  that  can 
ever  come  upon  the  bridge. 


Strain  upon  the  arch-suspension  rods. 

There  is  one  suspension  rod  If  diameter  at  each  arch,  and 
at  each  panel,  consequently  the  weight  on  each  portion  of 
the  bridge  corresponding  to  the  length  of  a  panel ;  that  is,  the 
weight  of  each  lineal  portion  of  9  feet  9  inches,  will  be  sus 
tained  by  4  rods,  having  a  united  cross-section  of  6  square  in 
ches.  The  greatest  weight  of  the  bridge  and  its  load  has 
been  found  to  be  3281  pounds  per  foot  lineal.  The  weight 
on  one  panel  will  therefore  be  31,989  pounds,  and  the  tension 
per  square  inch  =  5331  pounds,  or  about  one-tenth  of  the 
breaking  weight. 

The  arches  and  suspension  rods  are,  therefore,  more  than 
sufficient  to  sustain  the  greatest  load  that  can  ever  come  upon 
the  bridge,  without  any  assistance  from  the  truss. 


Strain  upon  the  counter-bracing  produced  ly  the  action  of  the 

arch. 

This  strain  will  be  estimated  by  supposing  one  half  of  the 
bridge  to  be  loaded  with  one  ton  per  foot  lineal,  which  is  not 
counterpoised  by  any  weight  on  the  opposite  side. 


RAILROAD   VIADUCT.  177 


Let  the  weight  on  the  half  span  be  supposed  concentrated 
at  the  centre  of  gravity  Gr. 

B  Q  =  one-fourth  span  =  37J  feet. 

0  a  =  f  P  S  (nearly)  =  16      " 

A  a  =  ^  A  O2  +  0  a2  =  ^1122  +  162  =        113     " 
G-  d  =  i  A  a  =  28J   « 

fg    =  \IPS-  IP  S=&P  £  (nearly)  =    12     « 
gh   =2fg=  24      « 

The  weight  concentrated  at  Gr,  is  supposed  to  be  160,000 
pounds. 

TP"  .^     Q.  $ 

The  resultant  in  the  direction  Gr  A,  will  be  — 7i~n —  ~ 

Cr  C/ 

160,000  x  28J 

-j-7—      -  =  282,500  pounds. 

This  force  with  its  equal  at  A,  produces  an  upward  action 
upon  the  arch  at  /,  which  may  be  supposed  to  be  resisted  by 
the  application  of  a  force  at  this  point. 

The  required  force  at  /,  can  be  determined  from  the  pro 
portion  :  d  g  :  gf  : :  282,500  :  half  required  force  — 

282,500  x  24 

—£Q-    -  =  121,070  pounds. 

This  is  the  whole  amount  of  force  which  is  exerted  upon 
the  arches  of  both  trusses,  and  which  must  be  resisted  by  the 
counter-bracing.  The  estimate  is  only  an  approximation,  but 
it  is  considerably  on  the  side  of  safety ;  for  it  is  impossible  that 
the  weight  should  ever  be  concentrated  at  any  point  Gr,  and 
the  effect  of  the  portion  upon  P  Gr  is  to  assist  in  keeping  down 
the  arch.  It  is  not  necessary  for  practical  purposes  to  make 
an  exact  mathematical  calculation  of  this  strain:  it  is  suffi 
cient  to  obtain  a  near  approximation,  with  the  assurance  that 
all  errors  are  on  the  side  of  stability.  From  these  considera 
tions,  it  appears  that  the  upward  force  upon  the  arch  by  the 
12 


178  BRIDGE   CONSRTUCTION. 

action  of  the  load  upon  the  opposite  side,  is  not  more  than 
121,070  pounds,  or  62,000  pounds  to  each  truss. 

As  there  are  twelve  panels  between  A  Gr,  there  are  conse 
quently  twelve  counter-braces  to  resist  this  force,  and  if  each 
of  them  sustained  an  equal  portion,  there  would  only  be  5170 
pounds  to  each;  but  a  more  nearly  correct  distribution  of  the 
pressure  is,  to  allow  nothing  for  the  strains  at  A  and  6r?  and 
double  the  average  for  the  strain  at/;  consequently,  the  great 
est  possible  strain  upon  any  counter-brace,  would  be  less  than 
11,000  pounds,  or  only  262  pounds  per  square  inch. 

If  iron  rods  had  been  used  for  counter-bracing,  a  cross- 
section  of  1  square  inch  to  each  panel  would  have  been  an 
ample  allowance. 

We  will  now7  estimate, 

2nd.     The  strength  of  the  truss  itself  without  the  arch. 

DATA. 

The  distance  of  the  centre  of  gravity  from  the 
point  of  support,  is  39  feet. 

The  distance  from  middle  of  upper  to  middle 
of  lower  chord,  is  17  " 

The  resisting  cross-section  of  upper  chord,  is        205  sq.  in. 

The  combination-keys  of  the  lower  chord  cut 
off  one-half  inch  on  each  side  of  each  chord  plank, 
or  three  inches.  The  splice  at  each  panel  4J  in 
ches  ;  the  combination-bolt  about  equivalent  to  1 J 
inches ;  there  remains,  therefore,  for  the  actual  re 
sisting  area  of  the  lower  chords  135  " 

We  will  assume  that  timber  should  never  be  subjected  to 
the  action  of  a  weight  that  could  be  sufficient  to  impair  the 
elasticity ;  and  that  within  the  elastic  limits,  the  resistances  to 
compression  and  extension  are  equal.  We  will  also  leave  out 
of  view  the  additional  strength  which  is  derived  from  the  con 
tinuity  of  the  spans ;  since  this  advantage  would  not  be  pos 
sessed  by  the  spans  at  the  extremities,  or  by  an  isolated  span 
of  the  same  extent.  The  calculation,  therefore,  will  be  made, 


RAILROAD  VIADUCT.  170 

fts  such  calculations  always  should  be  made,  under  the  most 
unfavorable  circumstances. 

As  each  upper  chord  presents  a  resisting  area  of  205  square 
inches,  and  each  lower  chord  an  area  of  135  square  inches, 
and  as  the  resistances  per  square  inch  are  supposed  to  be  equal, 
the  position  of  the  neutral  axis  will  not  be  in  the  middle,  but 
must  be  determined  by  the  condition  that  the  moments  of  the 
resistances,  or  the  products  obtained  by  multiplying  each  area 
by  the  distance  from  the  neutral  axis,  shall  be  equal. 

If  x  represent  the  distance  of  the  neutral  axis  from  the  mid 
dle  of  the  lower  chord,  17  —  x  will  be  its  distance  from  the 
middle  of  the  upper  chord. 

Let  R  represent  the  average  strain  per  square  inch  upon 
the  lower  chord ;  as  the  strains  are  in  proportion  to  the  dis 
tance  from  the  neutral  axis,  the  strain  per  square  inch  on  the 

17 x 

upper  chord  will  be  expressed  by  R  -      — . 

x 

The  equation  of  equilibrium  will  be 

R  ^-^X  205  X  (17  —  a;)  =  135  Rx 

x 

70  x2  —  6970  =  59245 
x  =9-4 


X 

The  equation  of  moments  will  be 

262-500 
7-6  x  -808  R  x  205  +  R  x  135  x  9-4  =  39  x  —^- 

2527  R  =  5118750 

R  =  2000  pounds  per  square  inch  (nearly). 
It  appears,  therefore,  that  the  strain  upon  the  bottom  chords, 
on  the  supposition  that  the  arches  are  omitted,  and  the  bridge 
loaded  with  a  train  of  locomotives  producing  a  weight  of  one 
ton  per  lineal  foot,  would  be  2000  pounds  per  square  inch. 

Strain  upon  the  ties. 

It  has  been  shown  that  the  greatest  strain  upon  the  ties  at 
the  middle  of  a  bridge  is  equal  to  the  greatest  load  that  can 


180  BRIDGE   CONSTRUCTION. 

ever  come  upon  one  panel ;  consequently,  it  will  be  the  same 
as  was  determined  for  the  weight  upon  the  arch  suspension 
rods,  or  31,989  pounds. 

There  are  4  rods  at  each  panel,  2  to  each  truss. 

The  rods  at  the  middle  of  the  bridge  are  1J  inches  diameter. 
The  united  cross-section  of  the  4  rods  will  be  7  square  inches, 
and  the  strain  per  square  inch  4,569  pounds. 

The  end  rods  sustain  the  weight  of  one-half  the  bridge  and 
its  load.  Continuing  the  same  hypothesis  as  formerly,  the 
weight  of  the  half-span  and  its  half-load  has  been  found  to  be 
262,500  pounds. 

This  is  sustained  by  4  rods,  each  If  diameter,  the  united 
cross-section  of  which  will  be  9-6  square  inches.  And  the 
strain  therefore  27,344  Ibs.,  or  one-half  the  breaking  weight. 

The  pressure  upon  the  braces  will  bear  to  the  strain  upon 
the  rods,  the  proportion  of  the  diagonal  of  the  panels  to  the 
perpendicular ;  this  proportion  is,  in  the  present  case,  as  19  : 
16.  We  have  therefore 

31,989  x  19 
For  the  pressure  on  the  middle  braces ^ = 

38,000  pounds  nearly. 

The  cross-section  of  the  braces  in  the  middle  is  168  square 
inches. 

The  pressure  per  square  inch  on  the  middle  brace  is  226 
pounds. 

For  the  pressure  upon  the  end  braces,  we  have 

262,500  x  19 

-T7g =  311,718  pounds. 

The  2  trusses  at  the  ends  contain  6  braces,  4  of  which  are 
of  pine,  6x7;  the  others  are  of  oak,  7x7. 

The  united  cross-section  will  therefore  be  266  square  inches. 

The  pressure  per  square  inch  will  be  1172  pounds. 

It  is  necessary  to  inquire  whether  this  pressure  of  1172 
pounds  per  square  inch  will  cause  the  brace  to  yield  by 
flexure. 

The  braces  at  the  ends  are  three  in  number,  placed  side  by 
side,  and  supported  in  the  middle  by  the  counter-brace.  Two 
cases  present  themselves  for  consideration. 


RAILROAD  VIADUCT.  181 

ls£.  Flexure  may  take  place  in  the  direction  of  the  plane 
of  the  truss ;  in  which  case  the  resistance  will  be  due  to  2 
braces  6  inches  deep,  7  inches  broad,  and  9J-  feet  long ;  1 
brace  7  inches  deep,  7  inches  broad,  and  9J  feet  long. 

2nd.  Flexure  may  take  place  in  a  direction  perpendicular 
to  the  plane  of  the  truss,  in  which  case  the  resistance  will  be 
due  to  2  braces  7  inches  deep,  6  inches  wide,  and  19  feet 
long ;  1  brace  7  inches  deep,  6  inches  wide,  and  19  feet  long. 

The  formula  which  expresses  the  extreme  limit  of  the  re 
sistance  to  flexure  when  the  material  is  white  pine,  is 

9000  BI>3  . 

W  = j-z i*1  which  /  is  in  feet,  and  the  other  dimen 
sions  in  inches. 

By  substituting  the  proper  dimensions,  we  have  in  the  first 
case 

9000  x  7  x  63  x  ^ 
For  the  two  6  x  7  pieces,  W= &&9~      ~  =  301>561 

9000  x  7  x  73 
For  the  one  7  X  7  piece,  W= Q  .9  239,434 


Total,  representing  the  extreme  limit  of  resistance,      540,995 
The  actual  strain  upon  the  braces  at  the  end  of  one  truss 

is  155,876  pounds,  which  is  sufficiently  far  below  the  limit  to 

insure  perfect  security  against  flexure  in  the  first  case. 

For  the  second  case,  in  which  flexure  is  supposed  to  take 

place  in  a  direction  perpendicular  to  the  plane  of  the  truss, 

we  have 

9000  X  fi  x  73  v  2 
For  the  two  6x7  pieces,  W=-        ~-J$r^~^  =  102,600 

9000  x  7  x  73 
For  the  one  7  x  7  piece,  W=  -    -  1Q2  59,850 


Total,  representing  the  extreme  limit  of  resistance,  162,450 
This  calculation,  it  must  be  remembered,  is  based  on  the 
supposition  that  the  truss  is  not  assisted  by  arches,  and  that 
its  load,  independently  of  the  weight  of  the  structure,  consists 
of  a  train  of  locomotives  of  the  largest  class,  extending  entirely 
across  the  span.  These  are  hypotheses  which  will  never  ex 
press  the  actual  condition  of  the  Susquehanna  bridge ;  but  it 


182  BRIDGE   CONSTRUCTION. 

is  proper  to  examine  them,  as  bridges  are  frequently  built  on 
similar  plans  without  arches,  and  on  roads  over  which  very 
heavy  trains  are  carried. 

The  result  of  the  calculation  proves  that,  in  the  first  case, 
where  the  counter-brace  forms  an  intermediate  support,  and 
reduces  the  length  of  the  unsupported  portions  of  the  braces  to 
9J  feet,  flexure  cannot  take  place  in  the  plane  of  the  truss. 

In  the  second  place,  where  there  is  no  intermediate  sup 
port,  and  the  three  braces  are  supposed  to  yield  laterally  in  a 
direction  perpendicular  to  the  plane  of  the  truss,  the  resistance 
is  not  sufficient,  and  flexure  might  take  place  under  the  hypo 
theses  assumed,  unless,  by  the  addition  of  keys  and  bolts,  the 
three  braces  are  made  to  act  as  one  piece,  in  which  case  the 
formula  will  give 


W=  =  1,386,000  nearly, 

which  is  more  than  double  the  stiffness  in  the  first  case,  and 
9  times  as  great  as  the  maximum  strain. 

The  conclusion,  therefore,  is,  that  in  a  truss  constructed 
upon  these  principles,  but  without  arches,  it  is  highly  impor 
tant  that  the  braces  at  the  ends  of  the  spans  should  be  stiffened 
laterally  by  bolts  and  keys. 


Strain  upon  the  floor  beams. 

The  floor  beams  are  7  X  14,  placed  3  feet  8  inches  from 
centre  to  centre ;  the  interval  between  chords  is  15J  feet ;  the 
greatest  weight  upon  any  floor  beam  would  be  equivalent  to 
4J  tons  applied  at  the  centre. 

On  the  supposition  that  the  deflection  is  ^  inch  to  1  foot, 

^AgL--   7*143        -640* 

•0125 1*~  -0125  x(15-5)2~ 

The  actual  weight  is  9000. 

The  deflection  produced  by  this  weight  would  be 

C402  :  9000  : :  TTJ  •  ~rx  or  JAA  °f  an  inc^  Per  f°ot  in  length. 
The  actual  deflection  caused  by  the  passage  of  a  locomo- 


RAILROAD    VIADUCT.  183 

tive,  allowing  6  tons  weight  upon  a  pair  of  driving  -wheels, 

1-4  217 

will  be  T      x  15'5  =  -,  or  one  half  inch  nearly. 


To  determine  the  strain  upon  the  fibres,  we  must  use  the 

formula  11  =  0  ,  ,2  in  which  all  the  dimensions  are  in  inches, 
J,  o  ct 

and  R  expresses  the  maximum  strain  per  square  inch, 

3  x  9000  x  15-5  x  12 

U  =  -  o  ^  n  ^  -M3  -  ~  1830  Pounds  per  square  inch, 

Zt  X    I    X   l'± 

The  effect  of  the  track  strings  in  distributing  the  pressure 
over  several  adjacent  floor  beams,  has  not  been  taken  into  con 
sideration,  because  it  is  not  safe  to  make  any  allowance  in  or 
dinary  cases.  As  a  locomotive  of  the  first  class  occupies  con 
siderable  space  in  the  direction  of  the  track,  several  beams  may 
be  loaded  with  equal  weight  at  the  same  time,  and  one  could 
not  assist  another  ;  besides,  there  must  be  joints  in  the  track 
strings,  and  at  the  joints  the  beam  must  bear  the  whole  strain. 

This  weakness  is  compensated  in  the  Susquehanna  Bridge  by 
making  the  floor  beams  at  the  joints  two  inches  wider  than  at 
the  intermediate  points  ;  the  strain  upon  them  will  therefore 
be  less  than  that  previously  determined,  in  the  proportion  of 
7  to  9  :  it  will  consequently  be  1424  pounds  per  square  inch. 


Strain  upon  the,  counter-brace*. 

It  has  been  shown,  that  when  the  load  upon  a  bridge  is 
uniform,  the  counter-braces  do  not  act.  Their  office  is  to  re 
sist  the  upward  action,  produced  by  an  unequal  distribution 
of  the  weight.  The  greatest  variable  load  is  estimated  at  one 
ton  per  lineal  foot,  or  160,000  pounds  to  the  half  span ;  the 
effect  of  which  is  resisted  by  the  counter-braces  of  the  opposite 
side.  The  conclusion  arrived  at  in  considering  the  subject  of 
the  action  of  the  counter-braces,  was,  that  the  greatest  resist 
ance  which  it  was  ever  necessary  for  any  one  of  them  to  op 
pose,  was  equal  to  the  pressure  upon  the  braces  of  the  middle 
panel  caused  by  the  action  of  the  greatest  variable  load.  The 
greatest  load  upon  one  panel  is  ten  tons,  or  20,000  pounds ; 


184  BRIDGE   CONSTRUCTION. 

20,000  x  18 
the  strain  m  the  direction  of  the  brace  is ^ —  - 

36,000.    The  cross-section  of  the  2  braces  is  84  square  inches. 
The  pressure,  430  pounds  per  square  inch. 

Lateral  braces. 

The  greatest  strain  upon  the  lateral  bracing  of  a  bridge, 
would  be  that  caused  by  the  action  of  the  wind  in  a  violent 
tornado.  It  is  probable  that  this  force  is  far  greater  than  it  is 
usually  estimated.  The  observations  of  the  writer  at  the  Sus- 
quehanna  Bridge,  dnring  the  tornado  which  caused  the  loss  of 
six  of  the  unfinished  spans,  led  him  to  believe  that  the  direct 
effect  of  the  storm  was  increased  by  reflection  from  the  surface 
of  the  water.  It  appears  reasonable  to  suppose  that  if  the 
direction  of  the  wind  is  such  as  to  strike  the  surface  of  the 
water  at  an  angle  of  reflection,  it  must  be  thrown  upwards, 
and  its  eifect  would  be  to  augment  the  pressure  upon  any  sur 
face  exposed  to  its  action.  In  covered  bridges  particularly,  it 
is  probable  that  this  reflected  current  acting  against  the  under 
side,  and  in  opposition  to  gravity,  might  so  reduce  the  weight 
of  the  structure  as  to  cause  it  to  be  blown  off  the  piers.  The 
possibility  of  this  contingency  we  propose  to  examine  after  the 
direct  effects  have  been  considered.  In  the  absence  of  positive 
information,  the  necessary  data  will  be  assumed. 

The  Pennsylvania  Railroad  Viaduct  is  designed  to  be  left 
entirely  open.  The  amount  of  side  surface  in  the  chords  and 
braces  is  688  square  feet  to  each,  and  as  the  wind  can  act  upon 
both  trusses,  the  surface  presented  will  be  1376  square  feet. 

There  are  two  sets  of  lateral  braces,  one  at  the  lower,  the 
other  at  the  upper  chord.  As  the  upper  set  is  subjected  to  a 
greater  strain  than  the  lower,  the  calculation  will  be  limited 
to  it.  The  upper  set  of  lateral  braces  may  be  considered  as 
resisting  the  force  of  the  wind  on  one-half  the  side  surface  of 
the  truss  (688  square  feet),  and  the  force  of  the  roadway  and 
railing  (6  X  160  =  960  square  feet).  Allow  for  obliquity  of 
direction,  which  would  increase  the  surface  352,  making  a 
total  of  2000  square  feet. 


RAILROAD  VIADUCT.  185 

If  we  suppose  that  a  storm  could  bo  so  violent  as  to  cause 
a  pressure  of  30  pounds  per  square  foot,  the  whole  lateral  force 
would  be  60,000  pounds.  The  force  upon  the  lateral  brace 
rods  would  be,  at  each  end,  30,000  pounds ;  and  as  these  rods 
have  a  cross-section  of  1J  square  inches,  the  strain  per  square 
inch  would  be  24,000  pounds,  or  about  one-half  the  breaking 
strain. 

This  estimate  is  probably  beyond  the  extreme  limit  of  per 
fect  safety ;  for  the  force  of  the  wind  has  been  rated  twice  as 
great  as  is  given  in  tables  of  violent  storms.  No  allowance 
has  been  made  for  the  assistance  of  the  flooring  boards,  which 
is  very  considerable,  and  the  diagonal  braces,  which  transfer 
a  considerable  portion  of  the  pressure  to  the  lower  chords  and 
the  lower  diagonal  braces. 

The  strain  upon  the  lateral  braces  will  be  to  the  strain 
upon  the  rods,  in  the  proportion  of  the  diagonal  of  the  panel 
to  the  perpendicular,  or  as  18  : 15J- ;  it  will  therefore  be  36,000 
pounds,  or  1200  pounds  per  square  inch. 

The  lateral  braces  5  X  6,  18  feet  long.  The  limit  of  re 
sistance  to  flexure,  as  determined  by  the  formula  -  — ^ — 

will  be  29,988  pounds.  The  strain  under  the  present  hypo 
thesis,  is  36,000  pounds ;  consequently,  the  lateral  braces 
would  require  support  in  the  middle. 

If  supported  at  the  middle  point,  the  length  of  the  unsup 
ported  portions  would  be  reduced  to  9  feet,  and  the  limit  of  re 
sistance  would  be  119,952  pounds. 

In  which  case  there  would  be  a  considerable  surplus  of 
strength. 

It  appears,  therefore,  that  for  security  in  a  very  violent  tor 
nado,  the  lateral  braces  should  be  supported  in  the  middle  of 
their  length,  which  is  the  case  in  the  Susquehanna  Bridge. 

Strain  upon  the  diagonal  braces. 

If  the  bridge  is  in  a  perpendicular  position,  the  strain  upon 
the  diagonal  braces  will  result  from  the  force  of  winds  upon 
the  side  trusses.  In  the  Susquehanna  Bridge,  as  the  sides  are 
open,  and  there  is  a  close  parapet  6  feet  high  on  the  top  of  the 


186  BRIDGE   CONSTRUCTION. 

bridge,  the  centre  of  gravity  of  the  surface  will  be  very  high, 
and  may  be  taken  at  the  level  of  the  top  chord. 

Estimating  the  side  surface  at  2000  square  feet,  and  the 
force  of  a  storm  at  30  pounds  per  square  foot,  the  total  force 
will  be  60,000  pounds  horizontally. 

The  strain  in  the  direction  of  the  diagonal  will  be  to  this 
horizontal  strain  in  the  proportion  of  23  to  15,  it  will  conse 
quently  be  92,000  pounds. 

The  diagonal  braces  are  5x6,  and  23  feet  long. 

If  they  are  bolted  together  at  their  intersections,  the  resist 
ance  in  the  direction  of  a  plane  passing  through  them  will  be 
considerably  greater  than  in  a  perpendicular.  The  estimate 
should  of  course  be  made  in  the  direction  of  least  resistance. 

The  least  lateral  resistance,  when  the  two  braces  are  bolted 
together,  will  be  twice  that  of  a  post  23  feet  long,  6  inches 
broad,  and  5  inches  deep.  The  limit  of  resistance,  as  deter  - 

9000  b  d3 
mined  by  the  formula  W—  --  j~2  -  ?  w^  be 

pounds 


nearly  ;  as  this  is  the  force  that  will  actually  cause  the  brace 
to  yield  by  flexure,  or  the  extreme  limit  of  resistance,  it  will 
not  be  safe  to  allow  more  than  8000  pounds  to  each  brace,  if' 
it  acts  singly,  or  16,000  pounds  if  each  pair  is  bolted  ;  and  as 
the  force  to  be  resisted  is  92,000  pounds,  there  will  be  required 
in  the  first  case  12  pairs  of  diagonal  braces,  and  in  the  second 
case  6  pairs. 

It  appears,  therefore,  that  a  large  amount  of  diagonal  bra 
cing  is  necessary  to  resist  a  strain  capable  of  producing  a  pres 
sure  of  30  pounds  per  square  foot.  These  braces  cannot  be 
permanently  introduced  until  after  the  arches  are  in  place,  and 
the  loss  of  the  six  spans  at  the  Susquehanna  Bridge  was  in 
consequence  of  the  unfinished  condition  of  the  bridge,  which 
did  not  admit  of  the  permanent  introduction  of  a  sufficient 
number  to  resist  the  effects  of  the  sudden  and  violent  tornado 
to  which  it  was  exposed. 


RAILROAD  VIADUCT.  187 


Resistance  to  sliding  upon  the  supports. 

A  bridge  which  is  securely  braced  diagonally,  laterally  and 
vertically,  may  yield  to  the  force  of  the  wind  by  sliding  upon 
the  tops  of  the  piers,  or  wall-plates.  The  possibility  of  failure 
from  this  cause,  in  the  case  of  the  Susquehanna  Bridge,  we 
will  proceed  to  examine. 

Assume  that  the  force  of  the  wind  is  80  pounds  per  square 
foot,  and  that  an  additional  force  is  exerted  by  reflection,  equal 
to  10  pounds  per  square  foot,  acting  vertically  in  opposition  to 
gravity,  and  reducing  to  this  extent  the  weight  of  the  bridge. 
These  conditions  will  probably  require  a  greater  power  of  re 
sistance  than  will  ever  be  actually  necessary  in  service. 

The  experiments  of  the  writer  on  the  friction  of  wood  upon 
wood,  and  of  wood  upon  stone,  give  T7T>  of  the  pressure,  as  the 
least  resistance  to  sliding. 

The  width  of  the  floor  of  the  bridge  being  24  feet,  and  the 
span  160  feet,  the  number  of  superficial  feet  of  surface  will  be 
3840,  and  the  upward  force  of  the  wind,  at  10  pounds  per 
square  foot,  will  be  38,400  pounds. 

The  weight  of  one  span  without  a  load  is  205,000  pounds ; 
deducting  the  first  result,  the  difference,  which  is  the  resisting 
weight  of  the  span,  will  be  167,400  pounds ;  T72  of  this  weight 
will  give  for  the  friction,  or  resistance  to  sliding,  97,650  pounds. 

Estimating  the  maximum  amount  of  surface  in  the  open 
bridge  at  2500  square  feet,  and  the  force  of  wind  at  30  pounds, 
the  pressure  would  be  75,000  pounds,  or  22,650  pounds  less 
than  the  resistance. 

It  is  proper  to  conclude,  therefore,  that  the  bridge,  if  not 
weather-boarded  on  the  sides,  can  never  be  blown  over  by  a 
tornado,  provided  the  lateral  and  diagonal  braces  are  sufficient. 

Anchor-bolts  are  used  to  fasten  the  bridge  to  the  piers,  and 
in  addition  to  this,  the  masonry  has  been  extended  as  high  as 
the  top  chord.  Without  these  precautions  the  bridge  appears 
to  be  secure ;  with  them  no  storm  can  possibly  have  power  to 
move  it. 


188  BRIDGE   CONSTRUCTION. 

Power  of  resistance  of  the  Susquehanna  Bridge,  on  the 
supposition  that  the  arches  and  truss  form  but  a  single  system. 

The  strength  of  each  system  has  now  been  separately  ex 
amined,  and  calculations  made  of  the  strains  upon  every  part. 
It  remains  to  consider  their  action  when  united,  on  the  suppo 
sition  that  each  contributes  to  sustain  the  load  in  proportion 
to  its  powers  of  resistance.  This  hypothesis  does  not  express 
the  exact  conditions  of  the  problem,  but  it  is  the  only  one  that 
can  be  assumed,  and  the  deviation  from  the  truth  is  much  less 
than  some  engineers  suppose.  The  objection  which  is  urged 
against  a  combination  of  two  systems,  is,  that  either  one  or 
the  other  must  measurably  sustain  the  whole  weight,  and  the 
one  which  is  not  active,  is  merely  an  incumbrance  to  the  other. 
This  objection  is  plausible,  but  the  assumption  on  which  it  is 
based  is  contrary  to  truth.  One  system,  as  has  already  been 
stated  in  the  general  consideration  of  this  subject,  could  bear 
the  whole  weight  only  on  the  supposition  that  it  was  abso 
lutely  incompressible,  which  is  not  the  case.  It  is  very  clear, 
that  if  either  system  is  overloaded,  it  will  yield,  and  the  other 
will  be  brought  into  action. 

A  very  brief  consideration  of  this  case  will  be  sufficient. 

The  strain  upon  the  braces  at  the  middle  of  the  span,  will 
be  the  same  as  in  the  former  case.  As  the  roadway  is  on  top, 
and  the  weight  of  a  passing  load  is  transmitted  to  the  lower 
chords  and  arches  through  the  medium  of  the  braces,  the  latter 
must  be  capable  of  sustaining  the  weight  upon  one  panel. 

The  strain  in  this  case  has  already  been  found  to  be 

For  the  middle  braces,  226  pounds  per  square  inch. 

The  force  transmitted  by  the  braces  to  the  lower  chords  is 
not  resisted  by  the  ties  only,  but  also  by  the  arch-suspension 
rods,  the  united  cross-section  of  which  is  13  square  inches.  . 

The  weight  to  be  sustained  is  31,989  pounds. 

The  strain  per  square  inch  is     2,451 

The  vertical  force  at  the  ends  is  resisted  by  the  arch  and 
the  braces.  If  the  hypothenuse  of  the  skew-back  expresses 
the  pressure  in  the  direction  of  the  arch,  the  perpendicular  will 
represent  the  horizontal,  and  the  base  the  vertical  pressure,  aa 


RAILROAD   VIADUCT. 


189 


also  the  proportions  of  the  resisting  surfaces.  As  the  hypothe- 
mise  in  the  present  case  is  33,  and  the  base  15,  the  resisting 
surface  which  the  arches  are  capable  of  opposing  to  the  weight, 
will  be  15  X  9  X  4  =  540  square  inches. 

The  cross-section  of  the  braces  is  266  square  inches,  which 
expresses  the  resisting  surface  in  the  direction  of  the  diagonal, 

1  266  x  16 
and :rg =  portion,  which  opposes  a  vertical  resistance 

=  224  square  inches. 

The  whole  area  which  resists  the  pressure  at  the  ends  of 
the  bridge,  may  therefore  be  estimated  at  806  square  inches, 
and  as  the  weight  has  been  found  to  be  262,500  pounds,  the 
pressure  per  square  inch  at  the  ends  will  be  326  pounds. 

The  portion  of  this  weight  sustained  by  the  truss  will  be 
266  x  326  =  86,716  pounds,  to  resist  which  the  cross-section 
of  the  8  rods  is  16'6  square  inches,  and  the  strain  per  square 
inch  5,224  pounds. 


Estimate  of  the  longitudinal  strains. 

In  estimating  the  resistance  of  a  truss  composed  of  two 
systems,  it  is  not  correct  to  assume  that  it  will  be  equal  to  the 
sum  of  resistances,  which  each  separately  would  be  capable 
of  opposing,  because  the  strains  upon  the  several  portions  will 
depend  upon  their  distances  from  a  common  neutral  axis. 

Centre  of  resist 
ance  of  top 
chord. 

do.  of  arch. 

Neutral  axis. 

Centre  of  resist 
ance  of  bottom 
chord. 

do.  of  skew-back. 

If,  for  example,  we  consider  the  truss  A  B  without  the  arch, 
the  upper  and  lower  chords  sustain  nearly  equal  strains,  and 
the  neutral  axis  will  be  nearly  equidistant  from  both  ;  but  if  an 
arch  be  added,  having  a  centre  of  resistance  at  a  considerable 
distance  below  the  line  .#,  the  neutral  axis  will  be  brought 


190  BRIDGE   CONSTRUCTION. 

lower,  and  the  proportions  maybe  such  that  it  will  fall  exactly 
upon  the  lower  chord,  in  which  case  the  latter  will  sustain  no 
portion  whatever  of  the  strain ;  as  a  consequence,  the  resist 
ance  of  the  system  will  not  be  equal  to  the  sum  of  the  resist 
ances  of  its  component  parts. 

The  first  step  in  the  calculation  must  therefore  be,  to  de 
termine  the  areas  of  the  resisting  surfaces,  and  the  position  of 
the  neutral  axis. 

DATA. 

The  height  of  truss,  from  middle  of  top  to  middle  of  bottom 
chord,  is  17  feet. 

From  middle  of  bottom  chord  to  middle  of  skew-back,  5J 
feet. 

From  middle  of  top  chord  to  middle  of  arch  at  crown,  1J 
feet. 

Resisting  area  of  top  chord,  A,  410  square  inches. 

Resisting  area  of  bottom  chord,  B,  270  square  inches. 

Resisting  area  of  arch,  at  crown,  (7,  1044  square  inches. 

Resisting  area  of  perpendicular  of  skew-back,  D7  1051 
square  inches. 

Let  x  =  distance  of  neutral  axis  from  (7,  =  x. 

Dist.  of  A  from  neutral  axis,  x  +  1*25. 
"  B  "  IT  — (a; +1-25)  =  15-75  — a?. 

D  «  15-75  —  x  +  5-25  =  21  —  x. 

The  pressures  upon  the  resisting  surfaces  will  be  in  pro 
portion  to  their  distances  from  the  neutral  axis,  and  if  R  re 
presents  the  greatest  strain  per  square  inch  upon  the  resisting 
surface,  which  is  at  the  greatest  distance  from  the  axis,  which 
in  the  present  case  is  _Z>,  the  pressure  upon  the  other  surfaces 
will  be  per  square  inch. 

x  +  1-25 
Upon  the  resisting  surface  A         Rx  ~^r~^~ — 

15-75  _z 


RAILROAD  VIADUCT.  191 

The  resistance  of  each  surface  will  be 
A        =        AR- 


B        = 


The  equation  of  moments  will  be 

A  R  (*2r-iJ)  (x  +  1-25)  +  OR  (|j  -*)  x  =  B  R 
15*75 

(^-)  (is-75  -*)  +  !>£  (21-*), 

Whence  410  (x2  4-  2-5  x  +  1'56)  +  1044  x2  =  270  (248  — 
31-5  x  +  x2)  +  1051  (441—42  x  +  x). 

Reducing  133  x2  +  53,672  x  =  529,811 

x  =  9-75  feet,  very  nearly. 

Consequently  the  distance  of  the  neutral  axis  from  the 
middle  of  the  upper  chord,  will  be  11  feet. 

and  from  the  middle  of  the  lower,  06    " 

We  are  now  prepared  to  determine  the  strain  upon  each  of 
the  resisting  surfaces. 

Continuing  the  former  notation,  in  which  R  represents  the 
average  strain  per  square  inch  on  the  surface  .Z>,  the  strains 
upon  the  other  surfaces  will  be,  per  square  inch, 

At    A     =     R  x  j^|g     -     -977    R 

/> 

At    B     =     ^  =     *533    R 


At     0     =     Rs         =     '866    R 
At    D     =     - 


The  whole  strain  upon  each  surface  in  terms  of  72,  will  be 

Upon  A     =       410     x     -977     R     =       400    R 

"      B     =       270     x     -533     R     =       144    R 

"      0     =     1044     x     -866     R     -       904    72 

"      D     =     1051     x   1-000     R     =     1051    72 


192  BRIDGE    CONSTRUCTION. 

The  weight  upon  the  half  span  has  been  shown  to  be 
262,500  pounds. 

The  distance  of  the  centre  of  gravity  from  the  skew-back 
is  38  feet. 

The  moment  of  the  weight  will   therefore  be  262,500   X 
38  =  9,975,000. 
This  is  resisted  by 

400      R     x     11        =       4,400    R^ 
+       144      R     x       6         =          864    R  \ 
+      904      R     x       9-75    =       8,815    R  [  :     15>9C 
+    1,051      R     x     11-25   =     11,824     R  J 

The  equation  of  equilibrium  between  the  acting  and  resist" 
ing  forces  will  then  be 

15,909    R     =     9,975,000 

R     =        627  pounds. 

The  average  strain  per  square  inch,  upon  each  of  the  re» 
sisting  surfaces,  will  therefore  be, 

Upon  the  upper  chord  627     x     '977    =     613  pounds. 

"       lower     "  627     x     -533    =    334      « 

"       arch  at  crown,         627     x     -866     ==    543      " 
"  "      skew-back,  627     x          1    =  .  627      " 

The  calculations  for  the  Susquehanna  Viaduct  have  been 
extended  to  minute  details,  in  order  to  illustrate  the  principles 
of  calculation,  and  test  the  strength  of  every  part  of  this  im 
portant  structure. 

The  following  summary  of  results,  will  be  convenient  for 
reference. 


GENERAL   SUMMARY. 

No.  of  feet  B.  M.  white  pine,  in  one  span,  56,944     feet. 

«           «                  "      oak,             "  2,070      « 

"         cubic  feet  timber,  per  lineal  foot,  30-7   " 

Weight  of  timber  per  lineal  foot,  in  pounds,  1,105  pounds, 

"         cast-iron,  in  one  span,  '  10,551      " 

"               "         per  lineal  foot,  66      " 
a        bolts,  exclusive  of  arch-bolts  and 

nuts,  11  352      « 


RAILROAD   VIADUCT.  193 

VYeight  of  bolts,  exclusive  of  arch-bolts,  per 

lineal  foot,  71  pounds. 

"         arch  bolts,  3,568      « 

"         nuts,  90T      " 

"         finished  bridge,  per  lineal  foot,  1,281      " 

"         loaded  bridge,  «  3,281      " 

Cost  of  material  for  one  span,  $2,039  73 

"       workmanship,  for  one  span,      1,163  10 

Total  cost  for  one  span,  $3,202  83 

Cost  per  foot  lineal,  without  roof,  20  00 

If  the  arch  sustains   the  whole  weight,  the   pressure  at  the 
crown  will  be  453  Ibs.  sq.  in. 

If  the  arch  sustains  the  whole  weight, 

the  pressure  at  the  skew-back  will  be  450         " 

The  strain  on  suspension  rods,  5,331         " 

"          counter-brace,  262         " 

.Resisting  area  of  upper  chords,  410  sq.  in. 

"  "      lower  chords,  270       " 

If  the  arch  is  omitted,  the  strain  on  the 

lower  chords  loaded,  will  be  2,000  Ibs.  sq.  in* 

Strain  upon  the  ties,  in  the  middle,  4,569         " 

"         "         "      at  the  ends,  27,344         " 

"         "        braces,  in  the  middle,  226         " 

"         "  "      at  the  ends,  1,172         « 

"         "       counter-braces,  430         " 

"         "        floor  beams,  1,830         " 

Greatest  possible  strain  upon  lateral  braces,    1,200         " 
"  "         "         "      rods,  24,000         « 

"  "         "         "       diagonal   "  133         « 

No.  of  pairs  of  diagonals  required  for  one  span,  6. 

Power  of  wind  on  side  surface,  75,000  Ibs, 

Resistance  to  sliding,  97,650    " 


Strains  upon  the  parts  when  loth  systems  are  united. 

Upon  the  braces,  in  the  middle  of  the  span,  226  Ibs.  sq.  in* 

"        suspension  rods,  2,451          " 

"        braces  at  the  ends,  326          " 
13 


194  BRIDGE   CONSTRUCTION. 

Upon  the  suspension  rods  at  ends,  5224  Ibs.  sq.  in 

"        upper  chord,  613 

«        lower  chord,  334         « 

"        arch  in  centre,  543         " 

"          "    at  skew-back,  '  627         " 


COVE  RUN  VIADUCT.     (Plate  3.) 

This  design  was  prepared  for  a  bridge  across  Cove  Run, 
on  the  Pennsylvania  Railroad,  but  in  consequence  of  peculi 
arities  of  location,  another  plan  was  submitted ;  it  is  inserted 
here  in  consequence  of  its  simplicity. 


Description. 

The  span  is  50  feet  from  skew-back  to  skew-back. 

Width  from  out  to  out,  9  feet. 

Height  of  truss  from  out  to  out,  10  feet. 

Number  of  trusses,  2. 

The  upper  chord  is  a  single  timber,  12  x  12,  of  white  pine. 

The  posts  are  of  locust,  6x6,  supporting  the  upper  chord. 

The  arches  are  composed  of  rolled  plates ;  each  arch  con 
sists  of  8  plates,  2  X  f ,  with  a  space  in  the  middle  of  two  in 
ches,  the  upper  and  lower  portions  being  separated  by  blocks 
of  cast-iron.  The  lower  chord  is  of  rolled  iron,  and  is  designed 
not  to  resist  the  thrust  of  the  arch,  but  to  connect  the  system 
of  counter-bracing. 

The  lateral  braces  are  of  wood,  supported  by  angle-blocks 
of  cast-iron,  and  connected  by  rods  J  inch  in  diameter.  The 
counter-brace  rods  are  one  inch  diameter,  passing  through 
angle-blocks  on  the  upper  chord,  and  connected  with  the  lower 
chord  by  means  of  eyes  passing  around  the  lower  lateral  braco 
rods. 


COYE   RUN   YIADUCT.  195 

Bill  of  Materials  for  one  Span. 

Cast-iron.  Cubic  inches. 

40  blocks  between  arches,  each  7J  cubic  inches  300 

36  post  plates,  each  30  cubic  inches  1,080 

40  coupling  plates  for  arches,  each  20  cubic  inches  800 
20  angle-blocks,  for  counter-brace  rods,  each  14  cubic 

inches  280 
40  angle-blocks,  for  lateral  brace  rods,  each  46  cubic 

inches  1,840 

40  washers  for  lateral  bolts,  3  cubic  inches  120 
6  angle-blocks,  for  diagonal  brace  rods,  each  14  cubic 

inches  84 
6  washers  for  diagonal   brace   rods,  each  3  cubic 

inches  18 

4  skew-backs,  each  300  cubic  inches  1,200 

Total  cubic  inches  of  cast-iron     5,722 
Weight  in  pounds,  1431. 

Bill  of  Malleable  Iron. 
32  plates,  or  1728  lineal  feet  of  rolled  iron,  2  X  f 

for  arches  31,104 

20  counter-brace  rods  {  diameter,  12  feet  long,  88 

cubic  inches  1,760 

10  diagonal  tie  rods  1  inch  diameter,  14  feet  long, 

132  cubic  inches  1,320 

20  lateral  brace  rods  £  inch  diameter,  8J  feet  long, 

61  cubic  inches  1,220 

18  bolts  through  posts  1J  inch  diameter,  13  inches 

long,  16  cubic  inches  288 

4  skew-back  bolts  1J  inch  diameter,  16  inches  long, 

28  cubic  inches  112 

80  short  bolts  for  arches,  12  inch  by  -f  diameter,  5J 

cubic  inches  420 

4  tie  plates-  3  X  J,  50  feet  long,  900  cubic  inches         3,600 

152  nuts,  each  3  cubic  inches  456 

44  nuts,  each  9  cubic  inches  396 

Total  cubic  inches  40,676 

Weight  in  pounds,  10,169 


196  BRIDGE   CONSTRUCTION. 


Wood. 

2  top  chords  12  x  12,  54    feet  long,  B.  M.  =  1,296 

36  lateral  braces       4x5,     8J         "                "  =  510 

44  oak  cross-ties  8  x    8,  10           "                "  =  2,346 

22  locust  posts         6  x    7,  10          "              "  =  770 

4^922 
Weight,  14,766. 

Recapitulation. 

Weight  of  cast-iron  in  bridge,         (single  span,)       1,431  Ibs, 
"          malleable  iron  in  bridge,          "  10,169  " 

"          timber  including  cross-ties,  14,766  " 

Total  weight,    26,366 
Weight  per  foot  lineal  of  bridge  528  Ibs.,  or  J  ton  nearly, 

Calculation. 

The  action  of  the  truss  consists  in  counter-bracing  the  arch, 
it  has  of  itself  no  sustaining  power.  The  abutments  resist  the 
thrust. 

Allowing,  as  usual,  one  ton  per  foot  lineal  as  the  maximum 
load  on  a  single  track  railroad  bridge,  the  weight  upon  the 
half-arch  will  be  63,183  pounds. 

The  height  from  middle  of  skew-back,  to  middle  of  arch 
at  curve,  is  10  feet. 

The  distance  of  the  centre  of  gravity  from  the  abutment, 
12  feet. 

The  horizontal  strain  at  the  centre  will  be 


2T=  .,  75,819  pounds 

The  strain  at  the  skew-back  will  be 


...... 

101,900  pounds  nearly,  or  one-third  more 


than  at  the  middle  of  the  span. 


IRON  BRIDGE  ACROSS  HARFORD  RUN,  BALTIMORE.      197 

To  resist  this  we  have  the  four  arches,  the  united  section 
}f  which  will  be  24  square  inches,  or  4200  Ibs.  per  square 
inch,  being  nearly  one-fourteenth  of  the  crushing  weight. 


Estimate  of  cost  of  Cove  Run  Viaduct. 

1,431  Ibs.  castings  @  2J  cts.  $  35  77 

10,169    "    malleable  iron  @  3-J  cts.  356  91 
1,800  feet  pine  scantling  @  $15  per  M. 

(board  measure)  27  00 

2,400    «    oak  scantling  @  $20  per  M.  do.  48  00 

770   "    locust     "        @  $40       "  30  80 

Making  152  bolts,  at  an  average  of  15  cts.  22  80 

Workmanship,  52  feet  @  $6  per  foot  312  00 

Total  cost  $833  28 
Total  cost  per  foot  lineal,  $16  00. 


IRON  BRIDGE  ACROSS  HARFORD  RUN,  BALTIMORE. 
(Plate  5.) 

This  bridge  was  built  by  Daniel  Stone,  of  Massachusetts 
It  furnishes  a  good  illustration  of  the  application  of  the  princi 
ple  of  Howe's  truss  to  an  iron  bridge.  There  is  much  simpli 
city  in  the  arrangement  of  the  details. 


Description. 

This  bridge  is  32  feet  long  and  66  feet  wide.  It  is  sup- 
ported  by  5  trusses,  at  intervals  of  16J  feet  from  centre  to  cen 
tre.  The  two  outside  trusses  are  above  the  roadway,  serving 


198  BRIDGE   CONSTRUCTION. 

both  as  a  support  and  as  a  railing ;  the  3  intermediate  trusses 
are  entirely  below  the  roadway.  In  the  middle  of  the  bridge 
is  a  railway  track,  which  is  directly  over  the  middle  truss,  one 
rail  being  on  each  side,  and  the  pressure  is  distributed  over 
the  trusses  by  means  of  two  longitudinal  timbers  immediately 
below  the  transverse  floor  beams,  and  under  the  rails,  sus 
pended  by  bolts  from  each  floor  beam.  By  this  arrangement 
it  is  evident  that  the  whole  weight  upon  the  railroad  track  is 
supported  by  the  middle  truss. 

The  upper  chord  is  represented  in  the  details  on  the  plate ; 
it  consists  of  a  cast  piece  in  the  middle  and  rolled  plates  on 
each  side  connected  by  bolts ;  the  rolled  plates  are  without 
joints.  Beneath  the  chords  are  placed  angle-blocks  at  the  pro 
per  intervals  of  the  panels  upon  which  are  cast  grooves  to  fit 
the  plates  of  the  upper  chord.  The  vertical  rods  are  in  pairs 
passing  through  the  chords  and  angle-blocks ;  they  are  1J  in. 
in  diameter. 

The  lower  chord  consists  of  four  plates,  4  in.  x  J,  below 
which  the  suspension  rods  pass,  as  represented  in  the  plate. 

On  the  lower  side  of  the  chord  suspension  rods  pass  through 
a  wrought  iron  plate,  the  end  of  which  is  extended  to  a  suffi 
cient  length  to  give  room  for  two  holes  to  receive  the  hooked 
ends  of  the  lateral  brace  rods. 

The  braces  and  counter-braces  are  equal  in  size,  and  the 
cross-section  is  in  the  form  of  an  X. 

The  dimensions  of  the  flanges  are  only  J  inch  by  2J  inches, 
and  the  area  of  the  cross-section  1J-  square  inches. 

The  horizontal  lateral  bracing  is  by  means  of  rods.  They 
are  hooked  into  holes  in  the  projecting  edges  of  the  plates  at 
the  bottom  of  the  suspension  rods ;  the  4  rods  forming  the 
diagonals  of  each  panel  of  the  lateral  bracing  unite  at  the 
centre,  where  they  pass  through  a  ring  8  inches  in  diameter, 
and  are  secured  by  nuts  on  the  inside  of  the  ring,  which  fur 
nishes  the  means  of  lateral  adjustment. 

The  planking  consists  of  two  courses.  1st  course,  2  inch 
yellow  pine,  laid  longitudinally.  2d  course,  J  inch  white  oak, 
inclined  at  angle  of  45°. 


BRIDGE  ACROSS  HARFORD  RUN,  BALTIMORE.      199 
Bill  of  Materials  for  Bridge  over  Harford  Run. 

CAST-IRON. 

Each  truss  contains, 
32  lineal  feet  of  upper  chord  cross-section  6-| 

square  inches  =  2592  cubic  in* 

30  horizontal  counter-braces,  each  49  inches 

long,  cross-sections  lg  inches  =  1654      " 

8  posts   over   abutments,  36  inches   long, 

cross-sections  1|  inches  =    324      " 

22  angle-blocks,  each  12  cubic  inches  =    264      " 

2  rollers  at  ends  of  truss,  3  inches  in  di 
ameter,  8  inches  long  =    112      " 

Total  cubic  inches  of  cast-iron  in  one  truss  —  4946      " 
The  weight  of  which  at  -|  pound  per  cubic 

inch  is  =  1237  pounds. 

MALLEABLE  IRON  IN  ONE  TRUSS. 

32  lineal  feet  of  top  chords,  cross-section  3 

inches  =  1152  cubic  in. 

32  lineal  feet  of  bottom  chords,  cross-section 

6  inches  =  2304  cubic  in, 

13  plates  under  lower  chords  7A  X  6  X  f      =    439      " 
26  suspension  rods,  1J  inches  diameter,  45 

inches  long  =  1440      " 

52  nuts  for  suspension  rods,  2x2x1          =     208      " 


Total  malleable  iron  in  one  truss  =  5543      " 

Weight  in  pounds  =  1386  pounds. 
For  the  railroad  track  are  required, 
32  bolts,  40  inches  long,  f  inch  diameter,  to 

suspend  the  longitudinal  timbers          =     387  cubic  in, 
22  nuts  for  same,  |  X  1|-  x  1£  =      37      " 

424      " 

Weight     106  pounds. 


20'0  BEIDGE   CONSTRUCTION. 

Each  set  of  lateral  braces  requires, 

12  rods,  |  inch  in  diameter,  9|  feet  long  602  cubic  in 

3  rings,  8  inches  in  diameter,  2  X  |  108      " 

710     « 

Weight  =  178  pounds. 

Wood  for  the  whole  Bridge. 

44  floor  beams  of  yellow  pine,  6  x  12, 17  feet  long  B.  M.  4,488 

Floor  plank  do.  2  inches  "      4,356 

do.          oak,  IJ  inches  «      3,267 

64  lineal  feet  of  timber  under  rails,  10  X  10  "         533 

8  cross  beams  for  lateral  braces,  5  X  6, 16  feet 

long  "         860 

13,504 
The  weight  of  which  at  3  pounds  per  foot  B.  M.  =  40,512  Ibs. 

Recapitulation  of  Bill  of  Materials. 

Cast-iron  in  5  trusses,  1237  pounds  each  6,185  Ibs. 

Malleable  iron  in  5  trusses,  1386  pounds  each,  6,930 
Bolts  for  railroad  track  106 

do.    for  2  sets  of  lateral  braces,  178  pounds 

each  356 

7,392  Ibs. 
Weight  of  wood    40,512  " 


Total  weight  =  54,089  " 

Estimate  of  cost. 

6185  pounds  cast-iron  @  2|  cents  $139  16 

7392       "       malleable  iron  @  3  j  cents  240  24 

13504  feet  B.  M.  timber  and  plank,  average  $15 

per  M.  203  56 

Workmanship,  22  lineal  feet  @  $13  per  foot  416  00 

Total  cost  $997  96 


IRON  BRIDGE  ACROSS  HARFORD  RUN,  BALTIMORE.     201 

Estimate  of  Cost  of  a  Single  Trade  Bridge  similar  to  tht 
above,  with  2  trusses,  16  J  feet  from  centre  to  centre. 

2474  pounds  cast-iron  @  2J  cents  $  55  66 

27T2       "      malleable  iron  @  3J  cents  90  09 

284       "                 do.          for    lateral    braces  and 

track  @  3J  cents  9  23 

3400  feet  B.  M.  of  timber  @  $15  per  M.  51  00 

32  lineal  feet  workmanship  @  $6-|-  208  00 

Total  cost         $413  98 
Average  cost  per  foot,  $13. 


Calculation, 

The  middle  truss,  which  bears  the  weight  of  the  railroad 
track,  sustains  a  much  greater  load  than  either  of  the  others. 
As  the  length  of  the  bridge  is  not  sufficient  for  more  than  one 
locomotive,  it  will  be  assumed  that  the  greatest  load  would  be 
23  tons,  or  1533  pounds  per  foot. 

The  permanent  load  will  be  the  weight  of  the  truss  and 
16J  feet  of  roadway.     It  will  therefore  be, 
For  the  truss  itself, 

cast-iron  1,237  pounds 

malleable  iron  in  truss  1,386       " 

"  "      lateral  braces,  &c.  284       " 

Total        2,907       " 
For  the  roadway,  3400  feet  B.  M.  @  3  pounds 

per  foot  10,200       " 

Total  weight        13,107       " 
Or  410  pounds  per  foot  lineal. 
Add  for  the  weight  of  locomotive  46,000       " 

Total  weight  of  middle  truss  and  load  •-=  59,107       " 


202 


BRIDGE   CONSTRUCTION. 


Principle  of  Calculation. 


w 


The  weight  of  the  half  truss  A  B  is  supposed  concentrated 
at  the  centre  of  gravity  (7,  and  acts  with  a  leverage  A  D  equal 
to  one-fourth  the  span.  It  is  sustained  in  equilibrium  by  the 
horizontal  pressure  in  the  middle  of  the  chord,  acting  with  a 
leverage  equal  to  the  height  of  the  truss.  The  equation  of 


moments  is  therefore  H  '  x  7i  = 


8  W  S 

-r  or  H  x  j-?  , 


H  =  horizontal  strain  upon  the  chords, 

w  =  weight  on  half-span  =  30,000  Ibs.  nearly, 

s  =  span  =  30  feet, 

li  =  height  of  truss  =    3J  ft. 

Therefore  H  =  3°'QQ°  *  -  =  64,286  Ibs.  which  is  nearly 

4   X    Oijr 

the  same  for  the  upper  and  lower  chords.  To  resist  this  strain 
we  have  in  the  upper  chord  8J  square  inches,  and  as  the  strain 
is  compressive  it  resists  with  its  whole  area  —  the  strain  is  there 
fore  8000  Ibs.  per  square  inch  nearly. 

The  lower  chord  has  a  resisting  cross-section  equal  to  the 
whole  area  6  square  inches,  for  as  the  span  is  only  30  feet  it  is 
not  necessary  that  there  should  be  a  joint. 

The  strain  per  square  inch  will  therefore  be  10,712  Ibs. 

The  suspension  rods  next  to  the  abutments  sustain  one-half 
the  weight  of  the  bridge,  or  30,000  Ibs. 

The  cross-section  contains  2-J  square  inches. 

The  strain  per  square  inch  with  the  greatest  possible  load 
will  be  12,000  Ibs. 

The  strain  upon  the  braces  will  be  to  the  strain  upon  the 
ties  in  the  proportion  of  the  diagonal  of  the  panel  to  the  per- 


LITTLE   JUNIATA   BRIDGE.  203 

pendicular,  or  in  the  present  case,  as  50  is  to  42.  It  is  there 
fore  35,700  Ibs.  And  as  the  cross  section  of  the  two  braces 
2J  square  inches,  the  strain  per  square  inch  will  be  15,800  Ibs. 


Recapitulation. 

Strain  per  square  inch  on  upper  chord  —    8,000  Ibs. 

«  "  "       lower  chord  =  10,712  " 

"  "  "       end  suspension  rods  =  12,000  " 

"  "  "       braces  =15,800  " 

Maximum  load  on  middle  truss  1847  Ibs.  per  square  foot. 

Weight  that  would  make  the  greatest  strain,  10,000  Ibs.  per 
square  foot,  would  be  1169  Ibs.  per  foot. 

Breaking  weight,  if  good  malleable  iron,  60,000  Ibs.  per 
square  inch. 

Greatest  strain  on  bridge  —  T45  the  breaking  weight. 


LITTLE  JUNIATA  BRIDGE.     (Plate  6.) 
Description. 

The  span  of  this  bridge  is  60  feet.  Its  peculiarity  consists 
chiefly  in  the  manner  of  constructing  the  arches  and  the  ar 
rangements  of  the  details.  The  arches  are  made  of  iron  rails 
of  the  U  form,  such  as  are  frequently  used  for  railroad  tracks. 
Two  lines  of  these  rails  are  placed  base  to  base,  breaking 
joints  with  each  other,  and  between  them  is  a  cast  plate  with 
projections  on  the  top  and  bottom  to  fit  into  the  cavities  of  the 
rails,  effectually  preventing  any  lateral  separation.  The  cast 
pieces  are  of  sufficient  length  to  extend  from  post  to  post  of 
each  panel,  and  are  varied  in  size  from  the  middle  to  the  end, 
so  as  to  be  at  every  point  proportioned  in  cross  section  to  the 
strain  which  they  are  required  to  bear.  This  condition  ren- 


204  BRIDGE   CONSTRUCTION- 

ders  it  necessary  that  each  panel  should  have  a  plate  of  dif 
ferent  length  and  thickness  from  the  next,  but  as  the  plates 
are  of  a  very  simple  form,  and  only  five  patterns  are  required, 
the  expense  is  trifling  in  comparison  with  the  advantage  of 
varying  the  size  according  to  the  pressures ;  the  cast  plate  is  1 
inch  by  5  inches  in  the  middle  of  the  span,  and  1 J  by  5  inches 
at  the  ends. 

The  cross  section  of  the  castings,  including  the  pro 
jection  on  the  top  and  bottom,  is  in  the  middle    8,214  sq.  in, 
At  the  ends  10,714     " 

The  number  of  square  inches  in  each  rail  4,794     " 

The  entire  cross  section  of  each  arch  at  the  mid 
dle  is  17,802     " 
And  at  the  ends                                                       20,302     " 

The  rise  of  the  arch  from  under  side  at  skew-back  to  under 
side  at  crown  is  8  feet  9  inches. 

The  upper  chords  are  3  in  number,  each  '5  X  9,  placed  1J 
inches  apart  to  allow  the  diagonal  rods  to  pass  between.  The 
length  should  be  36  feet.  The  lower  chords  are  2  in  number 
6x9,  placed  5  inches  apart,  and  continuing  from  pier  to  pier 
without  joints. 

The  posts,  which  extend  from  the  upper  to  the  lower  chords, 
are  4x6  inches,  and  9  feet  4  inches  long  from  bottom  of  top 
chord  to  top  of  bottom  chord.  These  posts  are  in  pairs,  placed 
5  inches  apart,  to  admit  of  the  passage  of  the  arches  between 
them.  Between  each  pair  of  posts  and  above  the  arch  is  a 
third  post  5  X  8  in.  of  hard  stiff  wood,  as  white-oak,  or  locust, 
the  office  of  which  is  to  transmit  the  pressure  of  a  passing  load 
directly  to  the  arch.  The  smaller  posts  serve  to  connect  the 
system  of  counter-bracing,  give  great  lateral  stiffness  to  the 
arch,  and,  were  the  failure  of  the  arch  under  any  circum 
stances  possible,  they  would  form  the  posts  of  a  framed  truss 
sufficient  of  itself  to  sustain  any  ordinary  load.  There  are 
rods  in  the  direction  of  both  diagonals  of  the  panels,  and  each 
set  is  in  pairs. 

The  rods  which  extend  from  the  top  chords  towards  the 
centre,  constitute,  with  the  posts  just  described,  a  distinct  sys 
tem,  which  may  or  may  not  contribute  to  any  considerable 


LITTLE   JUNIATA   BRIDGE.  205 

extent  in  sustaining  the  load  according  to  the  adjustment  of 
the  arch.  In  the  absence  of  the  arch  they  would  bear  the 
whole  weight.  These  rods  in  the  first  and  second  panels  are 
1J-  inches  diameter,  and  in  the  6  middle  panels  1  inch  diameter. 

The  rods  in  the  direction  of  the  other  diagonals,  or  those 
which  beginning  at  the  top  chord  incline  towards  the  ends  of 
the  truss,  serve  as  counter-braces.  They  have  no  action  in 
sustaining  a  direct  load  uniformly  distributed ;  should  the  truss 
settle  they  would  bend,  but  they  have  a  most  important  action 
in  resisting  any  upward  pressure,  such  as  is  produced  by  a 
weight  upon  one  side  of  the  truss,  when  not  counterbalanced 
by  a  corresponding  weight  on  the  opposite  side.  These  rods 
are  also  in  pairs,  but  they  are  only  -|  inch  in  diameter,  except 
in  the  last  panel,  where  there  is  a  single  rod,  the  diameter  of 
which  is  1J  inches.  Plates  of  annealed  copper  J  inch  thick 
are  placed  in  the  arches  between  the  ends  of  the  castings  to 
equalize  the  pressure  on  the  joints. 

The  width  of  truss  from  out  to  out  of  chord  is  19  feet,  and 
height  10  feet  10  inches. 

Between  chords  in  clear  16  ft.  1  inch. 

The  width  of  the  panels  from  middle  of  truss  is  6  ft.  1  inch. 

The  floor  beams  are  6  x  12  and  24  feet  long;  between 
supports  the  distance  is  16  feet  1  inch ;  they  are  placed  3  feet 
from  centre  to  centre,  and  the  weight  is  distributed  by  track 
strings  10  x  10  placed  under  each  rail.  These  string-pieces 
are  without  joints.  On  the  track  strings  are  cross-ties  of  white 
oak,  or  locust,  2  feet  apart  from  centre  to  centre,  4x6  inches 
in  cross-section,  laid  flat  side  down. 

There  are  4  panels  of  lateral  braces  to  each  span. 

This  system  of  lateral  braces  consists  of  diagonal  timbers 
5x7  resting  against  angle  blocks,  and  connected  by  1  inch 
bolts  extending  through  both  trusses.  The  same  arrangement 
is  used  for  both  top  and  bottom  chords.  The  system  of  diago 
nal  braces  is  represented  in  the  plan.  There  are  two  pairs  on 
each  pier,  and  three  piers  intermediate. 

The  length  of  the  arch  in  the  middle  being  63  feet  8  inches, 
the  most  convenient  length  for  the  iron  rails  would  be  21  feet 
3  inches ;  by  cutting  one  bar,  but  not  in  the  middle,  there  will 


206  BRIDGE   CONSTRUCTION. 

be  pieces  to  break  joints  at  the  ends.  The  half  of  21  feet  3 
inches,  or  10  feet  7-J  inches,  would  be  a  convenient  length  for 
the  castings,  and  they  would  all  be  of  uniform  length.  The 
reason  for  not  cutting  the  rail  at  the  middle  point  is  to  prevent 
any  of  the  joints  from  coming  opposite  to  each  other.  One- 
third  of  the  length  from  the  end  would  be  a  suitable  point  of 
division. 

Bill  of  Materials  for  one  Span. 
PINE. 

12  Upper  chords     5x9  37   ft.  long  B.  M.  1655 

4  Lower  do.          6x9  66  "  "  1188 

36  Small  posts        4x6  9J  "  "  684 

23  Floor  beams       6   x   12  24^  "  "  3312 

2  Track  strings  10    X  10  66  "  "  1100 

6  Purlines              5   X    10  22  "  "  550 

6        do.                 4  x     6  22  "  "  264 

12        do.                 3x4  22  "  «  264 

2  Guardrails       10    x    12  66  «  «  1320 

16  Lateral  braces    5x7  24  «  «  1120 

10  Diagonal  do.      5x6  13  "  "  1950 

1300  feet  B.  M.  white  pine  boards  1300 

Total  pine  =  14707 

WHITE   OAK. 

8  Pier  posts  8x8  9J-  feet  long 

50  Lineal  feet  posts  5x8  8  " 

34  Cross  ties  4x6  8  " . 

8  Bolsters  8x9  8  " 

Total  white  oak  B.  M.  =  1500 

CAST-IRON. 

18  bottom  angle-blocks     wt.  each     12  Ibs.  216 

18  top  "  "  25    "  450 

4  castings  for  top  chord,     "  50    "  200 


LITTLE   JUNIATA  BRIDGE.  207 

63  ft.  8  in.  of  iron  on  each  side       1808  Ibs.  3616 

4  skew-backs                 wt.  each       75     "  300 

36  post  plates  4x6i  in.     «               5     "  )  Rolled  platcg  180 

36        do.       13x6J"      "            20     "   V  would  be  720 

16        do.         8  x  8J-  "      "            17     "  j  Preferable-  272 

20  lateral  angle-blocks         "             13     "  260 

20  washers  for  lateral  bolts  "               3     "  60 

208         do.        small     do.    "                |  "  174 

Total  weight  of  cast-iron  6448 


Malleable  Iron  for  One  Span. 

63  feet  8  inches  of  iron  rail  in  each  arch,  cross- 

section  9-588  square  inches         1832  =  3864  pounds. 

16  diagonal  rods          1  J  in.  diameter,  13  ft.  long  868  " 

24         do.  1    "        «        13      "      827  " 

4  counter-brace  rods  1J  "         "         13       «      217  " 

32  do.  f  "         "         13      "      432  " 

10  lateral  bolts  1    "         "        19  ft.  6  in.  517  " 

60  short  bolts  through  chords  f  in.  diam.,  20  in. 

long  149  " 

44  short  bolts  through  posts    f        "         15  " 

long  82  « 

22  short  bolts  through  top  angle-blocks,  1  in.  di 

ameter,  9  in.  long  44  " 


Total  weight         7000 


Nuts. 

20  for  li  inch  bolts                    1T%  Ibs.  28  Ibs. 

66    "   1           "                             I    «  58  " 

104   "     f         «                             i    «  52  " 

32    "     f          «                             i    «  16  " 

Total  154  « 

Weight  of  rails  on  bridge  2560  " 


208  BRIDGE   CONSTRUCTION. 


Summary. 

Weight  of  timber          per  foot  945  Ibs.  total  weight  56,724  Ibs, 

do.       cast-iron            "  107  "            "            6,448  " 

do.       malleable  iron  "  119  "            "            7,154  " 

do.       iron  rails           "  43  "            "            2,560  " 

do.       bridge  "      1214  "  "          72,886  " 


Estimate  of  Cost  of  One  Span  of  Little  Juniata  Bridge. 

14,707  B.  M.  white  pine  $15                    $220  60 

1,500      "      white  oak  20                          30  00 

6,448  Ibs.  castings  2J  cts.           161  20 

7,000    "    malleable  iron  3J-    "             245  00 

154   "    nuts  9      "               13  86 


Total  cost  of  material  §670  66 


Workmanship. 

Framing  and  raising  66  feet,  including  fitting  of 

arches,  at  $6J  per  lineal  foot  $429  00 

Making  20       1J  inch  bolts               @  45  cts.  9  00 

"        56       1          "                    @  25   "  14  00 

"      104         f        "                    @  10    "  10  40 

"        32         f        "                    @  10   «  3  20 

Total  cost  of  workmanship  $465  60 


Cost  of  material  per  foot 

do.      work,  including  bolts, 
Metal  roofing  " 

Painting 

Total  cost  per  lineal  foot         $21  94 


LITTLE   JUNIATA   BRIDGE.  209 

Data  for  Calculation. 

Span  60  feet. 

Versed  sine  8     "    9  in. 

Cross-section  of  cast  plate,  in  middle  of  arch        8-214  sq.  in. 

"  "  at  end  of  arch  10-714      " 

No.  of  square  inches  in  the  section  of  each  rail     4-794      " 
Cross-section  of  arch,  at  middle  17*802      " 

"  "          ends  20-502      " 

"          of  upper  chords,  5x9  each,          45-000      " 

"          "    lower       "       6  x  9     "  54-000      " 

Small  posts  4x6     "  24-000      « 

Middle  posts  5x8     "  40-000      " 

Width  of  bridge  from  out  to  out  of  chords         19  feet 

"  between  chords  in  clear  16    " 

"        panels  from  middle  to  middle  of  post    6    "    1  inch. 
Floor  beams  6  X  12,  24  feet  long,  3  feet  from  centre  to  centre. 
Track  strings  10  X  10,  without  joint. 
Cross-ties  4  X  6,  2  feet  apart  from  centre  to  centre. 
Radius  at  middle  of  arch  55-8  feet. 

Central  angle  65°. 

Length  of  middle  line  of  arch  63  feet  8  inches. 

Hypothenuse  of  skew-back  7-5 

Perpendicular  6-32 

Base  4-03 

Length  of  rails  for  arches  21  feet  3      " 

Length  of  cast-segments  10    "    7J    " 

Distance  from  bottom  of  top  chord  to  top 

of  bottom  chord  9    "    4      "• 

Height  from  out  to  out  of  chords  10    "  10      " 

Total  weight  of  bridge  without  load  72,886  Ibfr* 

Maximum  load  of  60  lineal  feet  120,000    " 

Weight  of  bridge  and  load  192,886    " 

Cross-section  of  upper  chords  270  square  inches. 

"  lower  chords  180  " 

Calculation  of  Strains. 

The  calculation  as  in  the  case  of  the  Susquehanna  bridge, 
will  be  made  on  the  hypotheses — 
14 


210  BRIDGE    CONSTRUCTION. 

1.  That  the  truss  without  the  arch  bears  the  whole  load. 

2.  That  the  arch  without  the  truss  bears  the  whole  load. 

3.  That  both  systems  act  as  one. 

FIRST   HYPOTHESIS. 

That  the  truss  without  the  arch  bears  the  whole  load. 

Strain  upon  the  Chords. 

To  find  the  position  of  the  neutral  axis,  the  following  data 
are  necessary. 

Cross-section  of  upper  chords,  270  square  inches. 
"  lower  chords,  180       "         " 

Distance  from  middle  of  lower  chords  to  middle  of  upper, 
10  feet  1  inch. 

Let  x  =  distance  of  neutral  axis  from  middle  of  the  top 
chord,  (10-08 — x)  =  distance  from  middle  of  bottom  chord. 

Let  P  —  greatest  pressure  per  square  inch  upon  the  top 
chord.  The  pressure  being  proportioned  to  the  distance  from 
the  neutral  axis,  the  pressure  per  square  inch  on  the  lower 

chord,  will  be  P  ( — -—      — ).    The  resistance  of  the  top  chord 
\c 

will  be  270  P.     The  resistance  of  the  bottom  chord,  will  be 

180  P  ( — ^).     The  weight  on  one-half  of  the  loaded 

x 

bridge  is  96,433  pounds.    The  centre  of  gravity  from  point  of 
support,  15  feet. 

The  equation  of  equilibrium  270  P  x  +  180  P  (-  — ) 

•C? 

=  96433  x  15.     Also,  270  P  x  =  180  P  (— —)  from  the 

OCt 

second  of  these  equations,  we  find  x  =  4-5  feet,  and  from  the 
first  P  =  595  pounds  =  the  pressure  upon  the  top  chord,  and 

5*55 
595  (T^A)  =  610  pounds  per  square  inch  =  strain  upon  the 

bottom  chord. 


LITTLE  JUNIATA  BRIDGE.  211 


Strain  upon  the  Posts. 

In  the  middle  of  the  bridge  the  strain  upon  the  posts  can 
not  exceed  the  greatest  load  upon  one  panel,  or  19,288  pounds; 
this  is  sustained  by  4  posts,  each  4x6,  having  a  united  cross- 
section  of  96  square  inches.  The  pressure  per  square  inch 
in  the  middle  of  the  span  will  therefore  be  200  pounds. 

At  the  ends  of  the  truss  it  is  proper  to  calculate  the  cross- 
section  of  the  posts  at  176  square  inches,  for  if  the  arches  are 
omitted,  the  spaces  between  the  small  posts  must  be  filled  up 
by  extending  the  end  posts  to  the  lower  chords. 

The  weight  at  one  end  of  the  bridge  is  9644  pounds 

and  the  pressure  per  square  inch,  is  550      " 

The  formula  for  the  resistance  to  flexure  of  the  posts  is 

9000  b  d3 
w  =      —j-2  ---  . 

We  have  4  posts  4x6  =  9  feet  4  inches  long. 

"        2     "     5  x  8  =  9    "   4         <•' 
The  weight  which  would  cause  the  second  to  yield,  is  ex- 

2  x  9000  x  8  x  53 
pressed  by  w  =  -  j^^  --  =  206640, 


4  x  9000  x  6  x43 
and  for  the  4  smaller  posts  w  =  -  7oT\l  --  =  158693. 


Limit  of  resistance  to  flexure  =  365,333  pounds. 
Greatest  weight  to  cause  flexure  =  96,433  pounds. 


Strain  upon  the  Ties. 

The  strain  upon  the  ties  will  be  to  the  pressure  upon  the 
posts,  in  the  proportion  of  the  diagonal  of  the  panel  to  the  per 
pendicular,  or  as  12J  is  to  10  nearly,  or  as  5  :  4,  consequently 


5 
the  strain  upon  the  middle  ties  will  be  -  —.  --  =  24110  Ibs. 

The  cross-section  of  the  rods  is  -T854  X  4  =  3-1416  square 
nches. 
The  strain  per  square  inch  -will  be  7680  pounds. 


£12  BRIDGE   CONSTRUCTION. 

The  end  ties  will  bear  -    —j—  -  =  120,554  pounds. 

The  cross-section  of  the  4  rods  is  5  inches. 
The  strain  per  sqaare  inch,  24,110. 


Lateral  and  Diagonal  Braces. 

The  amount  of  side  surface  is  so  small,  that  no  doubt  can 
be  entertained  of  the  sufficiency  of  these  parts. 

(See  Calculation  on  Susquehanna  Bridge.) 


Moor  Beams. 

Allowing  the  heaviest  locomotives  to  have  18  tons  on  6 
irivers,  and  the  space  on  the  rails  occupied  by  3  pair  of  drivers 
to  be  11  feet,  the  weight  may  be  considered  as  equally  distrib 
uted  on  5  floor  beams,  which  would  give  3f  tons  to  each. 

This  weight  acts  at  a  distance  of  2  feet  5  inches  from  the 
centre  of  the  beam,  and  as  the  floor  beams  are  16  feet  1  inch 
between  supports,  a  weight  of  3|  tons,  at  a  distance  of  5  feet 
7ir  inches  from  the  point  of  support,  will  be  equivalent  to  3| 

»»3 

-£—  applied  in  the  middle  =  2JJ  tons,  or  5066  pounds. 
To  this  must  be  added  one-half  of  the  weight  of  the  beam  it- 

94  v  fi  x  3 

self  -    — £—  ~  —  216  pounds,  and   the  total  weight  in  the 

middle  of  the  beam  will  be  5282  pounds. 

The  formula  for  the  strength  of  a  beam  supported  at  the 

18  wl 
ends,  is  ft  =  ~T~TT  "where  I  is  in  feet, 

18  x  5282  x  16 

Therefore,  It  = 77 — ^02 "  ==  1  <  60  pounds  =  maxi 
mum  strain  per  square  inch. 

The  deflection  caused  by  the  passage  of  a  locomotive  with 
18  tons  weight  upon  the  drivers,  will  be  deduced  from  the 


LITTLE   JUNIATA  BRIDGE.  213 


6x12 
equation  w  =  TQ^^  =  .Q125  x  16  a  =  324°  Pounds  weight, 

that  will  cause  a  deflection  of  ^  inch  to  1  foot,  or  Jg  =  |  of 
an  inch  in  16  feet. 

The  actual  weight  being  5282  pounds,  the  deflection  will 

5282 
be  in  proportion,  or  §  x  ^TA  =  '65  inch  deflection  caused  by 


the  passage  of  a  locomotive. 


Counter-Braces. 

The  greatest  possible  strain  upon  the  counter-braces,  being 
equal  to  the  strain  upon  the  braces  of  the  middle  panels  due 
to  the  variable  load,  will  be  1200  pounds. 

The  cross-section  of  the  4  rods  f  diameter  is  1J  square 
inches.  The  greatest  possible  strain  per  square  inch,  9600 
pounds. 


SECOND   HYPOTHESIS. 

Calculation  of  the  strength,  on  the  supposition  that  the  arch 
supports  the  whole  weight. 

The  span  of  the  arch  is  60  feet,  and  rise  8  feet  9  inches. 

The  weight  on  the  half  arch  being  96,443  pounds. 

Distance  of  centre  of  gravity  from  support,  15  feet. 

Cross-section  of  two  arches  in  middle,  35-6  square  inches. 

Cross-section  of  two  arches  at  ends,      40-6  do. 

w  =  P  =  pressure  per  square  inch,  we  will  have 

P  x  35-6  x  8-75  =  96443  x  15, 

whence  P  =  4644  pounds  —  strain  per  square  inch — middle 
of  arch. 

The  pressure  at  the  skew-back  is  to  the  pressure  at  the 
jrown  as  the  hypothenuse  is  to  the  perpendicular,  or  as  7*50 : 
6-32. 

But  the  cross-section  at  the  skew-back  is  also  increased  in 


214  BRIDGE   CONSTRUCTION. 

the  proportion  of  20-3  to  IT'S.     The  pressure  at  the  skeTT- 
back  will,  therefore,  be  per  square  inch 

7*50      17a8 
4644  x          X          =  4832  pounds. 


Strain  upon  the  Counter-Braces. 

(See  figure  used  in  calculating  Susquehanna  Bridge.) 

The  greatest  variable  load  on  one  half  the  bridge  is  60,000 
pounds. 

We  will  have  in  this  case  B  0  =15       feet. 

Oa  =     |  x  8-75  =  6-50    " 


A  a=    452  +  6-562—  45-5     " 
aD  —11-4     " 

fa  =4-9    « 

aB  =9-8     " 

60000  x  114 
Resultant  in  the  direction  GrA  -  g-^r  -    ==      104270, 

1  04^70  y  9*8 
and  --  "09.5      ~  ==  50000  nearly  =  maximum   limit  of  the 

upward  pressure  upon  the  arch. 

The  strain  per  square  inch  upon  the  counter-brace  rods 
of  one  panel  resulting  from  the  pressure  will  be  10,000  pounds 
nearly. 


THIRD    HYPOTHESIS. 

Calculation  of  the  strain,  on  the  supposition  that  both  sys 
tems  act  as  one.  As  the  arch  thrusts  against  a  skew-back 
placed  upon  and  between  the  bottom  chords,  it  is  important  to 
inquire  whether  the  whole  strain  is  sustained  by  the  lower 
chord,  or  whether  any  assistance  is  derived  from  the  masonry 
'tself. 

Where  the  roadway  of  a  bridge  is  placed  upon  the  bottom 
chord,  the  chords  generally  rest  upon  the  tops  of  the  abutments, 
and  if  the  arch  is  attached  to  the  chord,  as  in  the  plan  now 
under  consideration,  it  is  evident  that  the  latter  must  bear  the 


LITTLE   JUNIATA   BRIDGE.  215 

whole  strain,  and  no  assistance  whatever  can  be  derived  from 
the  resistance  of  the  masonry.  When  the  roadway  is  on  the 
top  chord,  the  masonry  is  usually  carried  up  to  the  level  of  the 
road  with  an  offset  for  the  bottom  chord  to  rest  upon.  And  it 
is  never  necessary  in  this  case  that  the  lower  chord  should 
bear  the  whole  strain.  By  placing  a  wall-plate  behind  the 
ends  of  the  lower  chords,  and  driving  wedges  between  it  and 
the  chords,  a  pressure  is  thrown  upon  the  abutment,  which 
takes  off  precisely  an  equal  amount  from  the  strain  upon  the 
chord.  The  assistance  to  be  derived  from  this  arrangement  is 
very  great,  and  should  never  be  neglected  where  circumstances 
admit  of  its  being  employed.  It  is  not  safe  to  depend  entirely 
upon  the  resistance  of  the  abutment,  or,  in  continuous  spans, 
upon  the  counterbalancing  thrust  of  one  arch  against  the  next, 
for  the  loss  of  one  span  in  this  case  would  insure  the  destruc 
tion  of  the  whole. 

It  is  seldom,  however,  that  when  one  span  of  a  bridge  is 
carried  away  the  next  to  it  is  loaded  with  much  more  than  its 
own  weight,  and,  consequently,  the  true  minimum  of  the  size 
of  the  lower  chords  should  be  such  as  would  render  it  more 
than  sufficient  to  sustain  the  tension  arising  from  the  weight 
of  the  bridge.  It  will  be  safe  to  depend  upon  the  mutual 
assistance  of  the  spans  and  of  the  abutments  to  sustain  the 
greater  proportion  of  the  thrust  arising  from  the  variable  load. 
In  the  present  case,  as  the  spans  are  so  short  that  the  lower 
chord  can  be  made  without  joints,  there  is  a  greater  resisting 
power  than  is  required  to  sustain  the  loaded  bridge,  since  we 
have  seen  that  the  strain  was  less  than  600  pounds  per  square 
inch. 

The  present  calculation  will  be  made  upon  the  supposition 
that  the  chords  are  keyed  at  the  ends  next  to  the  abutments, 
and  in  close  contact  over  the  pins,  which  is  equivalent  to 
ioubling  the  resisting  area.  The  following  are  the  data  for 
calculation  in  this  case : 

From  middle  of  upper  to  middle  of  lower  chord,  10  feet. 

From  middle  of  upper  chord  to  middle  of  arch,  8  feet. 

Middle  of  skew-back  and  middle  of  lower  chord  on  same 
iiorizontal  line. 


216  BRIDGE   CONSTRUCTION. 

As  the  arch  does  not  abut  against  the  masonry,  but  rests 
upon  and  between  the  lower  chords  at  the  skew-back,  the 
limits  of  resistance  in  a  horizontal  direction  will  be  the  resist 
ance  of  the  cross-section  of  the  chords,  for  if  the  thrust  should 
be  greater  than  this,  whatever  may  be  the  strength  of  the 
arch,  the  ends  of  the  chords  would  be  crushed.  Consequently, 
the  total  resistance  of  the  arch  and  lower  chords  on  the  line 
of  the  skew-back,  must  be  equal  to  twice  the  resistance  of  the 
cross-section  of  the  chords,  or  to  400  square  inches. 
•  After  making  allowances  for  bolt-holes,  &c.,  the  cross-section 
of  the  upper  chord  is  270  square  inches. 

The  cross-section  of  the  arch  is  35-6  square  inches,  and 
allowing  the  resistance  of  iron  to  be  ten  times  as  great  as  that 
of  wood,  the  equivalent  cross-section  would  be  356  square 
inches. 

Let  x  =  distance  of  neutral  axis  from  middle  of  top  chord ; 
(x  — -66)  =  distance  from  middle  of  arch  ;  (10  —  x)  =  distance 
from  middle  of  bottom  chord. 

Let  P  =  maximum  pressure  per  square  inch  on  top  chord. 

P 

-  (x  — *66)  =  pressure  per  square  inch  upon  arch. 
x 

P 

—  (10  —  x)  =  pressure  per  square  inch  on  bottom  chord. 

x 

The  equations  of  equilibrium  are, 

270  P  x  +  356  -  (x—  -66)2  =  400  —  (10— -z)2  and 
x  x 

2  x  400  *  (10  — .T)2  =  96443  x  15. 

From  the  first  equation  we  find  x  =  4*64. 
And  from  the  second  P  =  292  pounds. 
The  strain  per  square  inch  on  the  upper  chord  being  292 
pounds. 

292  X  4 
On  arch  at  crown  it  will  be  ~77gi —  x  10  =  2517  pounds. 

292  x  5-36 
On  the  lower  chord  it  will  be       ^^ —  337  pounds. 


LITTLE  JUNIATA   BRIDGE.  217 


Vertical  Pressure  upon  the  Arch  and  Posts. 

The  pressure  of  the  arch  in  the  direction  of  the  tangent  at 
the  skew-back  may  be  resolved  into  two  components,  one  hori 
zontal,  and  the  other  vertical ;  these  will  be  proportioned  to  the 
perpendicular  and  base  of  a  right-angled  triangle,  of  which 
the  face  of  the  skew-back  is  the  hjpothenuse;  and  if  the 
length  of  the  hjpothenuse  be  taken  to  represent  the  thrust  of 
the  arch,  the  base  will  represent  the  vertical  pressure  or  por 
tion  of  the  weight  sustained  at  that  point. 

The  section  of  the  arches  at  the  ends  is  40-6. 

The  hypothenuse  of  the  skew-back  7*50. 

The  base  of  skew-back  4-03. 

The  proportion  of  surface  which  resists  the  vertical  pressure 

— ff-xTr   -  =  21-8  of  iron,  equivalent  to  218  square  inches  of 

I    u(J 

wood.     The  cross-section  of  4  posts,  each  4  x  6  =  96  inches. 

Total  resisting  surface  314  square  inches. 

The  weight  being  96,443  pounds. 

The  pressure  per  square  inch  is  307       " 

In  the  middle,  the  maximum  pressure  upon  the  posts  can 
never  exceed  the  amount  previously  determined  as  due  to  the 
weight  upon  one  panel. 

The  posts  at  the  ends  containing  96  square  inches,  and  the 
pressure  per  square  inch  being  307  Ibs.,  the  proportion  of  the 
weight  sustained  by  the  truss  will  be  29,472  Ibs.,  which 
produces  a  pressure  in  the  direction  of  the  diagonal  rods 

29472  x  5 

—7—  -  =  36,840  Ibs.     As  the  cross-section  of  the  four  rods 

is  5  square  inches,  the  strain  per  square  inch  will  be  7,368  Ibs. 
The  strain  upon  the  counter-braces  will  be  the  same  as  in  the 
other  cases. 

G-eneral  Summary  of  Results. 

No.  of  feet  B.  M.  white-pine  in  one  span  14,707 

"          "         white-oak        "      "  1,500 


218  BRIDGE   CONSTRUCTION, 

No.  of  pounds  of  cast-iron  6,448 

"       "  rolled-iron  7,000 

"       "  nuts  154 

Weight  of  timber  per  lineal  foot  945 

"         cast-iron      "        "  10T 

"         rolled-iron   "         "  119 

"         nuts  "        "  43 

"         finished  bridge  per  lineal  foot  1,214 

"         bolts  for  arches  per  foot  115 

Cost  of  workmanship  of  one  span  §466  00 

"       material  "         "  §671  00 

Total  cost  per  lineal  foot  $22  00 

If  the  truss  be  supposed  to  bear  the  whole  load, 
The  pressure  upon  the  top  chord  will  be  595  pounds. 

"         "         "        bottom  chord  will  be  610       " 

"         "         "         posts  in  the  middle  of  span  200       " 
"         "         "  a  post  at  end  of  span  550       " 

"         "         "  the  ties  in  the  middle  7,680       " 

"         "         "  the  ties  at  the  ends  24,110       " 

Maximum  load  upon  the  floor-beam  3f  tons. 

Equivalent  weight  in  middle  5,282  pounds, 

Maximum  strain  per  square  inch  1,760        " 

Deflection  of  floor-beam  by  weight  of  locomotive      *65  inches. 
Greatest  possible  strain  per  square  inch,  coun 
ter-braces  9,600  pounds 
If  the  arches  bear  the  load,  the  strain  will  be 

in  the  middle  of  arch,  per  square  inch-  4,644       " 

At  the  ends  of  the  arch,  per  square  inch  4,832       " 

Upon  the  counter-braces  10,000        " 

If  arches  and  truss  act  together  as  one  system, 
The   maximum   pressure    of    top    chord   per 

square  inch  292       « 

The  maximum  pressure  of  bottom  chord  per 

square  inch  337       " 

The  maximum  pressure  of  the  arch  per  sq.  inch    2,517       " 

Pressure  per  square  inch  on  posts  at  end  of  span     307       " 

«          «          "          "          in  middle  200       " 

Pressure  on  ties  at  end  of  span  7,368       " 

"          "          middle  of  span  7,680       " 


SHERMAN'S  CREEK  BRIDGE.  219 


SHERMAN'S  CREEK  BRIDGE  — PENN.  CENTRAL 
RAILROAD.     (Plate  7.) 

This  structure,  in  the  general  appearance  of  the  elevation 
of  the  side-truss,  bears  some  resemblance  to  a  Burr  Bridge,  but 
it  possesses  several  peculiarities. 

1.  The  truss  is  double,  consisting  of  three  rows  of  top  and 
bottom  chords,  and  two  sets  of  posts  and  braces. 

2.  The  truss  is  counter-braced  by  inch  rods,  placed   be 
tween  the  braces,  and  running  in  nearly  a  parallel  direction. 
These   rods   pass   through  bolster-pieces,  placed   behind   the 
posts  on  the  top  and  bottom  chords. 

3.  The  panels  increase  in  width  from  the  ends  towards  the 
middle  of  the  spans.     The  first  panels  are  9  feet  1J  inches 
from  centre  to  centre  of  posts. 

The  middle  panels  12  feet  1J  inches. 

The  bridge  consists  of  2  spans,  each  148  feet  3  inches  from 
skew-back  to  skew-back,  or  154  feet  6  inches  from  middle  of 
pier  to  end  of  truss.  The  pier  is  3  feet  2  inches  on  top,  and 
6  feet  at  skew-backs. 

The  foundation  of  the  pier  presented  some  peculiarities  in 
its  mode  of  construction.  Great  difficulties  were  apprehended 
in  consequence  of  an  opinion,  based  upon  information  given 
by  residents  in  the  vicinity  of  the  work,  that  the  rock  was  at 
a  great  depth,  and  was  covered  by  a  deposit  consisting  of  the 
ruins  of  an  old  dam.  As  the  rock  could  not  be  reached  by 
sounding,  before  the  excavations  were  commenced,  in  conse 
quence  of  the  large  stones  which  were  scattered  through  the 
gravel,  it  was  concluded  to  make  use  of  a  crib,  consisting  of 
timbers  solidly  and  compactly  framed  together  without  leav 
ing  space  between.  The  timbers  of  one  course  lie  in  imme 
diate  contact  with  those  of  the  next,  and  the  whole  are  bolted 
together  with  iron  rods.  The  intention  was  to  make  use  of 
this  as  the  frame  of  a  coffer-dam,  if  it  was  found  possible  to 
reach  the  rock,  and  keep  out  the  water ;  if  not,  to  use  it  as  an 
ordinary  crib,  and  fill  it  with  rough  stones. 


220  BRIDGE   CONSTRUCTION. 

The  process  of  excavating  the  foundation  was  commenced 
with  a  horse-power  dredging-machine,  consisting  simply  of  a 
scoop,  capable  of  holding  about  6  cubic  feet,  with  handles  at 
front  and  rear  by  means  of  which  it  could  be  held  down  by 
four  or  eight  men.  The  point  of  the  scoop  was  shod  with 
iron,  and  armed  with  teeth,  a  chain  was  attached  to  the  point 
passing  round  a  windlass,  to  which  a  horse  was  attached. 
With  this  simple  apparatus,  the  bottom  was  excavated  to  the 
surface  of  the  rock  in  a  few  days. 

The  crib  was  sunk,  puddled  on  the  outside,  and  the  water 
bailed.  It  was  found  to  answer  effectually  as  a  coffer-dam. 
The  masonry  was  carried  in  regular  courses  to  the  surface  of 
the  water,  the  space  between  the  regular  masonry  and  the 
crib  filled  with  stones,  and  the  whole  grouted  perfectly  tight 
with  hydraulic  cement.  The  whole  expense  of  the  founda 
tion  was  $400  —  including  excavation  with  machine,  bailing, 
puddling,  and  grouting. 


Bill  of  Timber  for  one  Span  of  Sherman's  Creek  Bridge. 

3  wall-plates            8    x  16  18  feet  long  B.  M.       576 
20  chords                   6    x  13  36        "         "  4,680 
10      "                      8    x  13  36        "        "  3,120 
10      "                      8    x  10  36        "        "  2,400 
20      "                      6    x  10  36        "         "  3,600 
56  posts  yellow-pine  9    x  12  23        "         "  11,592 

4  king-posts             9    x  16  23        "         "  1,104 
15  floor-beams           8    x  14  18        «         "  2,520 

14  «                  7    x  14  18        "        "  2,058 
56  lateral  braces        4J-  X     7  8J      "         "  1,213 

3      «         "           4J  x    7  13        "        «  103 

30  roof-braces            4x5  17        "         "  850 

56  check-braces         9    x  20  3        «         «  2,520 

56          "                  9    x  23  3        «        "  2,898 

60  main-braces           6x9  19        «         «  5,130 

15  tie-beams              8    X  10  19        «         "  1,900 

Amount  carried  over      46,264 


SHERMAN'S  CREEK  BRIDGE.  221 

Amount  brought  forward  46,264 

3  purlines                 4x6         20  feet  long  B.  M.  320 

135  rafters                   3x5         10J       "         "  1,772 

15  roof  posts            4x5           3         "         "  75 
30  knee-braces           5x5           5         "         "  312 

16  track  stringers      8   X    10         20         "         "  2,133 
3300  feet  B.  M.  }  inch  sheeting  for  roof  3,300 

56  arch-pieces            9  x  11         25         "         "  11,550 
7000  feet  B.  M.  inch  boards,  for  weather-boarding,  20 

feet  long  7,000 

72,726 

Weight  per  lineal  foot,  1,416  pounds. 
No.  of  cubic  feet  per  foot  lineal,  40. 


Bill  of  Counter-Brace  Rods  for  one  Span. 

4  rods  for  1st  panels  each  24  ft.  3  in.  long  1  in.  diam.    97.    ft. 

4  "  2d      "        "  24  "2  "  1  "  97.     " 

4  "  3d      "        "  24  "8  "  1  "  98-7  " 

4  "  4th     "        "  25  "  0  "  1  "  100.    " 

4  "  5th     "        "  25  "2  "  1  "  100-7  " 

4  "  6th     "         "  25  "8  "  1  "  102-7  " 

4  "  7th  or  middle  pan.  26  "0  "  1  «  104-0  " 

Total  lineal  feet  700 
Weight  in  pounds  at  2T605<j  per  foot  =  1,855  pounds. 


Arch  Suspension  Rods  for  one  Span. 


If  inches  diameter. 


4  rods  each    6  feet  8  inches  ^ 

8         "         10    " 

8        "        13    « 

8         "         15  feet  6  inches 

8         "        17    «    2      " 

8         "        18    "    2      " 

Total  length  322  feet. 

Weight  at  4-^  Ibs.  per  foot,  1,590  pounds, 


222  BRIDGE    CONSTRUCTION. 

Lateral  Brace  Rods  for  one  Span. 
15  rods  each  16  feet  9  inches  long  1  inch  diameter  655  pounds, 

Small  Bolts  for  one  Span. 

60  bolts,  through  arches,  47  inches  long  1  inch  diam.  622  Ibs. 
60  bolts,  through  chords  and  posts,  34  inches  long  f 

inch  diam.  255   " 

30  roof-bolts  36  inches  long  f  inch  diam.  135   " 

224  spikes  for  braces  |-  pound  each  168   " 


Dimensions  and  Data  for  Calculation  of  Bridge  at  Sher 
man's  Greek. 

Span  at  skew-backs  148  ft.  3  in. 
Whole  length  of  truss  for  one  span  154  " 
Out  to  out  of  chords  20  " 
Middle  to  middle  of  chords  19  " 
Resisting  cross-section  of  upper  chords  400  sq.  in. 
Resisting  cross-section  of  6  lower  chords,  deduc 
tions  for  splice,  check-brace  and  bolt,  and  al 
lowing  for  scarf-key  280  sq.  in. 
Versed  sine  of  lower  arch  20  feet. 
Cross-section  of  8  arches  800  sq.  in. 
Span  148J,  and  rise  20,  will  give  radius  172-25  feet, 
And  172-25, 152-25,  and  74-125,  express  the  pro 
portion  of  the  hypothenuse,  perpendicular,  and 
base  of  skew-back. 

Hypothenuse  of  skew-back  covered  by  arches  18  inches, 

Perpendicular             "             "         "  16      " 

Base                           "             "         "  7-6  « 

Distance  from  skew-back  to  bottom  of  chord  4J    u 

"          "      middle  of  skew-back  to  middle  of 

chord  4  ft.  5  in. 

Width  from  out  to  out  of  chords  16  "  2  " 

"      between  chords  in  the  clear  11  " 

Distance  from  centre  to  centre  of  floor-beams  5J  feet. 


SHERMAN'S  CREEK  BRIDGE.  223 

Weight  of  one-half  span  complete  (77  feet)        120,000  Ibs. 
Distance  of  centre  of  gravity  from  point  of  support  37  feet. 
Weight  of  one-half  span  with  load  275,000  Ibs. 

Distance  between  shoulder  of  post  15J  feet. 


Calculation  of  Truss  without  the  Arches. 

Let  x  —  distance  of  neutral  axis  from  top  chord. 
19  —  x  =  distance  from  bottom  chord. 
P  —  pressure  per  square  inch  on  top  chord. 
p 
-  (19  —  x)  —  strain  per  square  inch  on  bottom  chord. 

400  Pa:  =280  —  (19  —  x)2. 

x  =  8-3  =  distance  from  top  chord. 

And  19  —  x  —  10-7  =  distance  from  bottom  chords. 

10-7 
400  P  x  8-3  +  280  P  x  -^  x  10-7  =  275,000  x  37. 

P  —  1532  Ibs.  =  pressure  per  square  inch  on  top  chord. 
And  1532  x  -^-^  =  1,975  Ibs.  =  strain  upon  bottom  chord. 

The  bottom  chords  derive  some  assistance  from  the  masonry, 
Dut  as  the  roadway  is  on  the  bottom  of  the  truss,  little  oppor 
tunity  is  given  for  wedging  the  lower  chords,  and  for  this 
reason  the  assistance  to  be  derived  from  this  service  is  not 
estimated. 


Ties  and  Braces. 

The  weight  upon  the  middle  panel  (12J  lineal  feet)  is 
45,000  Ibs.  To  resist  this  there  are  four  posts,  the  cross- 
section  of  each  being  72  square  inches,  or  the  united  cross- 
section  288,  equivalent  to  156  Ibs.  per  square  inch. 

The  distance  between  the  shoulders  of  the  posts  being  IfyJ 
feet,  and  the  width  of  the  middle  panel,  exclusive  of  posts, 
11|  feet,  the  diagonal  will  be  19-3. 

19'3 
The  strain  upon  the  diagonal  will  be  45,000  x          =  56,000 


224  BRIDGE    CONSTRUCTION. 

Ibs.,  which  divided  by  the  cross-section  of  the  four  braces,  wiL 

56,000 
make  the  pressure  per  square  inch     <>..,»     =  260  Ibs. 

The  expression  for  the  limit   of  the  resistance  to  flexure 

9,000  x9x63 


,  . 

w  =  4  x  -    —  jz  •  —  gives  the  present  case  w= 

46,000  pounds,  or  for 

The  four  braces  184,000  pounds. 

The  actual  pressure  56,000        " 

Difference  in  favor  of  stability  128,000       " 

The  end  ties  sustaining  the  weight  of  half  the  bridge,  will 
be  at  275,000  pounds,  the  cross-section  being  as  before  288 
square  inches,  the  strain  per  square  inch  will  be  955  pounds. 

The  width  of  the  end  panel  being  8  J  feet  exclusive  of  posts 
and  the  distance  between  the  shoulders  of  the  posts  being  as 
before  15J  feet.  The  diagonal  will  be  17*7  feet,  and  the  pres- 

275,000  x  17-7 
sure  in  the  direction  of  the  braces  -    —  T  r7r~    -  =  314,000 

JLO'O 

pounds  =  1451  pounds  per  square  inch. 

The  limit  of  the  resistance  to  flexure  for  the  4  braces  is  ex- 

9  000  x  9  X  63 
pressed  by  w  =  -    -^r^—   —  X  4  =  223,000  pounds. 

As  the  pressure  is  314,000  pounds,  it  appears  that  with  the 
assumed  weight  of  a  train  of  locomotives,  or  one  ton  per  lineal 
foot  besides  the  weight  of  the  structure,  the  end  braces  would 
yield  by  lateral  flexure  in  the  direction  of  the  plane  of  the 
truss  if  not  supported  in  the  middle. 

If  an  intermediate  support  be  used,  the  resistance  will  be 
quadrupled,  and  will  be  amply  sufficient. 

It  is  also  necessary  to  examine  whether  the  braces,  if  sup 
ported  in  the  middle  in  the  direction  of  the  plane  of  truss,  could 
yield  laterally  in  the  direction  of  the  perpendicular  to  this 
plane  ;  the  relative  resistance  in  the  two  cases,  are  as  6  x  9  3  : 
9  x  6  3,  or  as  9  :  4.  The  limit  in  this  case  would  therefore  be 

223,000  x  9 

--  -7  -  =  502,000  pounds,  which  is  more  than  the  pres 

sure  (314,000  pounds). 


SHERMAN'S  CREEK  BRIDGE.  225 

It  appears  therefore  from  this  calculation,  that  if  the  arches 
are  omitted,  the  end  braces  should  be  supported  in  the  middle 
by  diagonals  in  the  opposite  direction.  As  an  additional 
security,  the  depth  should  be  increased  to  9  inches.  In  the 
other  panels  they  should  diminish  gradually  to  the  middle  of 
the  span,  where  the  original  dimensions  are  sufficient. 


Floor  Beams. 

The  floor  beams  are  7  x  14  inches,  width  in  clear  between 
supports  11  feet,  distances  from  centre  to  centre  5J  feet. 

The  weight  on  the  drivers  of  a  locomotive  18  tons,  may  be 
considered  as  distributed  nearly  equally  over  3  floor  beams, 
which  will  give  6  tons  for  each  beam. 

6  X  3  -r-  5*5  =  3*3  tons  =  the  equivalent  weight  in  the  middle 
of  the  beam 

18  w  I       18  X  6600  x  11 

It  =  = = r-j~2 =  yoz  pounds  =  maximum 

o  d~  (  X  J.4 

strain  per  square  inch. 

Lateral  Braces. 

The  lateral  braces  are  4J  x  7  and  8  feet  long.  The  pre 
valent  winds  are  usually  in  a  direction  nearly  parallel  to  the 
axes  of  the  bridge,  so  that  its  exposure  is  not  great.  Assume 
as  the  basis  of  a  calculation  that  the  sides  are  closely  boarded 
20  feet  high,  and  that  the  perpendicular  force  of  wind  may  be 
15  pounds  per  square  foot,  the  whole  pressure  upon  one  span 
•will  be  45,000  pounds.  As  there  is  lateral  bracing  both  above 
and  below,  this  pressure  would  be  resisted  by  4  lateral  rods  1 
inch  diameter  =  3-14  square  inches,  or  3,344  pounds  per  square 
inch. 

The  proportional  strain  upon  the  lateral  braces  would  be 

45,000  x  8 

— g =  72,000,  to  resist  which,  are  4  braces  4J  X  7  =  126 

square  inches  =  571  pounds    per  square  inch.     The   bearing 
surface  at  the  joints  does  not  much  exceed  one-half  the  area 
15 


226  BRIDGE   CONSTRUCTION. 

of  the  cross-section,  consequently  the  actual  pressure  at  the 
joints  will  be  about  1,000  pounds. 

The  limit  of  flexure  of  the  4  braces,  is  expressed  by  w  - 
3,000  x  7  x4 


The  maximum  pressure  is  72,000  pounds. 

Difference  in  favor  of  stability,  128,000  pounds. 

The  lateral  braces  cannot  yield  either  by  crushing  or  bend 
ing,  and  are,  therefore,  amply  sufficient. 

Could  the  bridge,  if  not  loaded,  be  blown  away  f 

The  weight  of  one  span  has  been  found  to  be  240,000 
pounds. 

The  resistance  to  sliding  would  be  120,000  pounds. 

The  pressure  of  wind  45,000       " 

Difference  in  favor  of  stability       75,000       " 
Could  the  bridge  yield  to  the  force  of  the  wind  by  rotation 

around  the  outer  edge  of  the  chord? 

The  effect   of  the  wind,  45,000  pounds, 

acting  with  a  leverage  of  10  feet,  would  give 

for  the  disturbing  force  450,000  pounds. 

The    resistance  =  weight    of    bridge  x 

half-width  from  out  to  out  =  240,000  x  8  =  1,920,000       " 

Difference  in  favor  of  stability     1,470,000       " 


Strain  upon  the  Knee-Braces. 

This  is  the  first  case  in  which  it  has  been  necessary  to 
calculate  the  strain  upon  a  knee-brace.  The  cross-bracing  of 
the  other  bridges  upon  which  calculations  have  been  made 
having  been  in  the  direction  of  the  diagonals. 

The  case,  however,  presents  no  difficulty.  Let  A  C  B  D 
represent  the  cross-section.  The  effect  of  the  pressure  of  wind 
on  A  C  is  equivalent  to  half  that  pressure  applied  at  the  point 
A.  A  force  at  A  tends  to  produce  rotation  around  B  and  (7, 
which  may  be  resisted  by  a  brace  in  the  direction  of  the  diago 
nal  A  B. 


SHERMANS   CREEK   BRIDGE. 


227 


The  pressure  upon  the  brace  will  bear  to  the  force  at  A  the 
proportion  of  the  diagonal  to  the  side  A  D.  If  the  brace  be 
removed  the  pressure  must,  nevertheless,  still  continue,  and  if 
it  is  resisted  by  a  brace  e  /,  the  pressure  upon  e  f  will  be 
greater  than  upon  A  B  in  the  proportion  of  A  D  to  e  D,  be 
cause  D  is  a  fulcrum  and  A  D  and  e  D  the  leverages  of  the 
acting  and  resisting  forces.  If  efis  parallel  to  A  B,  which  is 
generally  a  very  favorable  direction,  the  length  ef  and  A  B 
will  be  in  proportion  to  the  distances  D  e  and  D  A,  and  may 
be  substituted  for  them.  In  the  present  case  the  force  of  wind, 
45,000  pounds,  acting  with  a  leverage  of  ten  feet,  will  give  its 
moment  450,000,  or  225,000  pounds  acting  at  a  distance  of  20 
feet.  The  length  of  the  diagonal  is^2 +162  =  25*6  feet. 

'  22500  x  25-6 
and  the  strain  in  the  direction  ot  the  diagonal ^ 

=  36,000  pounds. 

The  length  of  the  knee-braces  being  5  feet,  the  strain  upon 

25-6 

them  will  be  36,000  X  —  =  184,000  pounds.  This  is  resist 
ed  by  15  braces  (one  to  each  post).  The  cross-section  of  each 
is  25  square  inches,  but,  as  the  bearing  surface  of  the  joint 
does  not  extend  over  the  whole  surface  of  the  section,  the 
resisting  portion  will  be  reduced  to  15  square  inches.  The 

strain  per  square  inch  will  therefore  be  rrr — -TK  =  818  pounds 

lO  X  JLO 

For  the  resistance  to  flexure  of  the  15  braces,  w  - 

9000  x  5  x  53 

-  x  15  =  3,555,000,  or  about  20  times  the  pressure, 
o 


228  BRIDGE    CONSTRUCTION. 

The  strain  upon  the  bolts  at  D,  will  be  to  the  vertical  com 
ponent  at  A,  in  the  proportion  of  D  E  to  E  A,  or  as  5 :  (25'6 

A  D 

—  5).     The  vertical  component  at  A  =  22,500  —  ~r~n  ~ 

JL  L> 
1  C* 

22,500  x  gQ  =  17,000.     Hence,  the  strain  upon  the  15  bolts 

will  be  17,000  X  4  =  68,000,  or  4,533  pounds  to  each  bolt,  01' 
10,000  pounds  per  square  inch  if  the  bolts  are  f  inch  diameter. 


Pressure  upon  the  Arch. 

For  this  calculation  we  have,  from  the  table  of  data, 
Span,  148  feet. 

Distance  of  centre  of  gravity  from  abutment,  37  feet. 
Rise  of  arch,  20  feet. 

Proportion  of  hypothenuse,  base,  and  perpendicular  of  skew 
back  =  18  7'6  and  16. 

Cross-section  of  8  arches,  800  square  inches. 

"1  C* 

800  x  r-o  =  711  proportion  to  resist  horizontal  thrust   at 

skew-back. 

7-6 
800  x  —  =  338  square  inches  to  resist  vertical  pressure 

at  skew-back. 

The  weight  for  one-half  span  loaded,  is  275,000. 
800  x  20  x  P  =  275,000  x  37. 

P  =  448  =  pressure  per  square  inch,  on  arches  in  middle. 

The  resisting  cross-section  at  the  skew-backs  is  the  same  as 
at  the  crown. 

The  pressure  is  greater  in  proportion  of  the  hypothenuse 

18 

to  the  perpendicular;  it  will  therefore  be  448  X  ^  =  504  Ibs. 

The  arches  are  therefore  more  than  sufficient  to  sustain  the 
whole  weight. 

When  both  systems  act  as  one. 

The  data  required  to  determine  the  strains  upon  the  chords 
and  arches  are, 


SHERMAN'S  CREEK  BRIDGE.  229 

Distance  from  middle  of  upper  to  middle 
:>f  lower  chord  19  feet. 

Distance  from  middle  of  skew-back  to  mid 
dle  of  lower  chord  4i  " 

Distance  from  middle  of  top  chord  to  mid 
dle  of  arch  3-5     " 
Cross-section  of  upper  chords                           400  square  in. 

lower       "  280          « 

"  arch  at  crown  800          " 

"  "      skew-backs  711          " 

Let  x  =  distance  of  top  chord  from  neutral  axis. 
«   x  —  3*5  =  distance  of  arch  at  crown  from  neutral  axis 
«    19  —  x=  "         bottom  chord  "  " 

«   23-5  —  x  —         "         arch  at  skew-back    "          " 
"   P  —  pressure  per  square  inch,  on  top  chord. 

p 

«    —(x  —  3-5)  =          "  arch  at  crown. 

x  v  ' 

p 

«    _  (19  —  x)  =  "  bottom  chord. 

p 

"    —(23-5  —  x)  =  arch  at  skew-back,  hori 

zontally. 

The  equations  in  this  case  are, 

400  P  x  +  -  800  (x  —  3-5)2  +  -  280  (19  —  x)2  +  -  711 

2/  X  X 

(23-5  —  x)z -275,000  x  37, 

and  400  P  x  +  800  -  (x  —  3-5)2  —  280  -  (19  —  x)2  4-  711 


From  the  second  of  these  we  find  x  =  ll'S. 

Consequently  the  distance  of  the  neutral  axis  will  be, 
Below  top  chord  11*8  feet 

"      arch  8-3    " 

Above  bottom  chord  7-2    " 

"      skew-back  11 -7     " 

These  values  substituted  in  the  first  equation  will  give  P 
K  222,000  =  7,175,000  x  11-8,  or 
P  —  381  Ibs.  =  pressure  per  square  inch  on  top  chord. 


230  BRIDGE   CONSTRUCTION. 

o.o 

381  x  =pr7n  —  268  Ibs.  =  pressure  per  square  inch  on  arch 

at  top. 

7-2 
381  x  -TTTo  =  232  Ibs.  =  strain  per  square  inch  on  lower 

chord. 

381  x  rppg  =  377  Ibs.  =  strain  per   square   inch   on  per 
pendicular  of  arch  at  skew-back. 


Vertical  Pressure. 

Assuming  that  the  weight  sustained  by  each  system  will 
be  in  proportion  to  its  power  of  resistance,  the  greatest  weight 
that  the  truss  can  sustain  will  be  the  limit  of  flexure  of  the 
braces  in  the  end  panels.  This  has  already  been  found  to  be 
223,000  pounds,  which  will  be  produced  by  a  vertical  pressure 

of  -     '—=rp=—    -  —  200,000  pounds  :  this  is  the  extreme  limit 

of  the  power  of  resistance  of  the  end  braces. 

The  proportion  of  surface  at  the  skew-back  which  resists 
the  vertical  pressure  is  388  square  inches.  If  we  suppose  the 
vertical  pressure  on  the  base  of  the  skew-back  to  be  the  same 
per  square  inch  as  the  horizontal  pressure  upon  the  perpendi 
cular,  it  will  be  capable  of  resisting  180,830  pounds  ; 

this  deducted  from  the  whole  pressure,  275,000        " 

will  leave  for  the  portion  to  be  sustained  by 
the  braces  94-170        " 

which  is  below  the  resisting  power.  The  actual  limit  of  the 
resisting  power  of  the  arch  is  very  great,  but,  assuming  that  in 
practice  it  is  not  safe  to  exceed  1000  pounds  per  square  inch, 
the  proportions  of  the  weight  sustained  by  the  truss  and  arch 
would  be, 


For  the  truss  275,000  x  _  =  68,700  nearly, 


338,000 
And  for  the  arch  275,000  x        ^        =  206,100  nearly- 


SHERMAN'S  CREEK  BRIDGE.  231 

These  numbers  will  give  for  the  strain  per  square  inch  on 
the  arch,  — T.KO —  =  600  Ibs.  nearly. 

£>o  ITAA  ,.  ~lt7*t7 

For  the  end  braces      *    r 9fF~  =  360  Ibs.  nearly. 

It  has  been  stated  that  the  bridge  at  the  western  end  is 
sustained  by  an  abutment  pier — it  is  proper  to  examine 
whether  the  resistance  which  it  is  capable  of  opposing  is  suffi 
cient  to  counterbalance  the  thrust  of  the  arch,  on  the  supposi 
tion  that  it  should  bear  the  whole  of  the  load. 

The  dimensions  of  the  abutment  pier  are  given  in  the 
following  figure,  except  the  length,  which  may  be  taken  at 
16  feet. 


Middle  of  arch  3  feet) 
below  top  of  pier,     j 


We  will  examine  the  conditions  of  equilibrium  on  the  sup 
position  that  rotation  takes  place  around  the  point  B.  The 
disturbing  force  is  the  horizontal  component  of  the  thrust  of 
the  arch  =  358,750  Ibs.  acting  with  a  leverage  of  16 J  feet,  its 
moment  will  therefore  be  358,750  x  16J-  =  5,919,375. 

The  resistances  are, 

1.  The  weight  of  the  masonry  above  0  B  =  110  perches 
of  3,750  Ibs.  =  412,500  Ibs.     The  distance  of  centre  of  gravity 
from  B  is  5  feet,  the  moment  will  be  2,062,500. 

2.  The  adhesion  of  the  mortar,  estimating  it  at  50  Ibs.  per 
square  inch,  or  one-half  the  tabular  strength  of  hydraulic  ce 
ment,  will  be  on  a  surface  of  160  square  feet  =  1,152,000  Ibs., 
and  its  moment  with  a  leverage  of  5  feet  =  5,760,000  Ibs. 

3.  The  vertical  pressure  of  the  arch  itself,  275,000  Ibs., 


232  BRIDGE   CONSTRUCTION. 

acting  with  a  leverage  of  9  feet,  -will  give  a  moment  275,000  x 

9  =  2,475,000. 

The  sum  of  the  moments  of  the  resisting  forces  will  be  2,062,500 

5,760,000 
2,475,000 

Total  10,297,500 
Moment  of  disturbing  force     5,919,375 


Difference  in  favor  of  stability  =  4,378,125 
As  this  difference  is  less  than  the  adhesion  of  the  mortar, 
it  appears  that  an  abutment  pier  of  dry  masonry  of  the  same 
dimensions  would  be  overturned. 

It  has  been  supposed  in  this  calculation  that  the  arch  bears 
the  whole  weight,  and  that  the  abutment  resists  the  whole 
thrust.  The  actual  horizontal  thrust,  with  the  two  systems 
acting  together,  was  found  to  be  377  X  711  ==  268,047.  The 
moment  will  be  268,047  x  16}  =  4,422,775.  The  resistance, 
omitting  the  strength  of  the  mortar  =  4,537,500.  From  which 
it  appears  that  if  we  disregard  the  adhesion  of  the  mortar,  the 
system  as  a  whole  would  be  very  nearly  in  a  state  of  equi 
librium,  the  difference  being  in  favor  of  stability.  The  prac 
tice  of  the  writer  in  proportioning  abutments  on  rock  founda 
tions  is,  to  disregard  the  adhesion  of  the  mortar,  throwing  this, 
whatever  it  may  be,  in  favor  of  stability ;  there  is  so  little  uni 
formity  in  the  strength  of  mortar,  and  so  much  liability  to 
cracks  occasioned  by  jars,  when  partially  set,  that  it  is  not  safe 
to  depend  too  mugh  upon  it.  If  the  proportions  and  weight 
of  an  abutment  prevent  it  from  overturning,  without  taking  the 
strength  of  the  mortar  into  consideration,  it  is  too  weak. 

When  the  base  is  to  any  extent  compressible,  it  is  not  suffi 
cient  that  the  disturbing  and  resisting  forces  should  be  in  a 
state  of  equilibrium,  a  condition  which  requires  the  resultant 
of  all  the  forces  to  pass  through  the  point  of  rotation.  But  it 
is  proper  that  the  resultant  should  pass  through  the  middle  of 
the  base.* 

*  This  calculation  was  made  before  the  completion  of  the  bridge ;  the 
correctness  of  the  conclusions  was  soon  confirmed;  the  pier  began  to 
crack  after  the  opening  of  the  road,  and  an  increase  of  thickness  by  the 
addition  of  buttresses  was  found  necessary. 


SHERMAN'S  CREEK  BRIDGE.  233 


Estimate  of  Cost  of  One  Span. 

72,726  feet  B.  M.  timber  12J  cts.  $909  07 

5,280  Ibs.  rolled  iron  3J    "  184  80 

3,300  square  feet,  roof  10      "  330  00 

Making  147  bolts  30      «  4410 

«         90    •«  10     «  9  00 

Workmanship  of  154  feet  at  8      "  12  32 


$2,708  97 
Cost  per  foot,  $18  24. 

Summary. 

Span  148  ft.  3  in, 

Width  of  pier  on  top  3  «  2  " 

"             "        skew-back  6  " 

Timber  in  one  span  72,726  " 

Weight  of  timber  per  lineal  foot  1,416  pounds, 

No.  of  cubic  feet  per  foot  lineal  40       " 

Weight  of  iron  in  one  span  5,280       u 

Width  from  out  to  out  of  chords  20  feet. 

"          middle  to  middle  of  chorda  19     " 

Versed  sine  of  lower  arch  20     " 

Radius  172,55     " 

Weight  of  half-span  loaded  275,000  pounds, 

Strain  upon  floor  beams  per  square  inch  902       " 
"           lateral  brace-rods  per  square  inch    3,444       " 

"          lateral  braces  571       " 

"           knee-braces  per  square  inch  818       " 

Pressure  per  square  inch  on  top  chord  381       " 

"             "             "       arch  at  crown  268       " 

"             "             "       lower  chord  232       " 
"             "             "       arch  at  skew-back       600       " 

"             "             "       end-braces  360       " 

"             "             "       middle  braces  260       " 


234  BEIDGE   CONSTRUCTION. 

RIDER'S  PATENT  IRON  BRIDGE.     (Plate  8.) 
Description. 

The  truss  of  the  Rider  bridge  is  principally  composed  of  an 
upper  and  lower  chord,  upright  posts,  and  diagonal  ties. 
The  upper  chord  is  made  of  cast  iron,  with  heavy  horizontal 
flanges.  The  lower  chord  is  made  of  wrought-iron.  The 
diagonal  ties,  or  suspension  rods,  are  also  made  of  wrought- 
iron,  and  are  secured  only  to  the  upper  and  lower  chord,  at 
regular  intervals,  running  upwards  and  downwards  diagonally 
with  the  chords,  and  at  nearly  right  angles  with  each  other. 
The  posts  are  of  cast-iron,  and  are  placed  at  equal  distances 
apart  along  the  whole  length  of  the  chords,  to  keep  the  upper 
and  lower  chords  asunder,  and  at  the  same  time  to  assist  in 
preserving  the  truss  in  line. 

A  wedge  is  inserted  on  the  top  of  each  iron  post,  under  the 
top  chord,  by  the  action  of  which  the  diagonal  rods  are  kept 
in  a  state  of  tension. 

Bill  of  Materials  for  a  Single  Span  of  60  feet. 

Height  of  truss,  7  feet.  Width  in  the  clear  of  chords,  12 
feet.  Floor-beams  of  wood. 

Cast-Iron. 

54  cast-iron  posts  6  feet  long,  cross-section  9  sq.  in.  8,784  Ibs. 
132  lineal  feet  cast-iron  top  chord,       "       15      "     5,940    " 
110         "         'caps  for  lower  chord,    "         2J    "        825    " 

Total  cast-iron  15,549    " 

Malleable  Iron. 
104  diagonal  ties,  each  9-2  ft.  long,  cross-section  f  §  4,500  Ibs. 


RIDERS   PATENT   IRON   BRIDGE.  235 

10  lateral  rods  each  17'7  long  by  1  inch  diam.  234  Ibs. 
104  small  bolts  for  top  chord  each  6  inches  long  by 

finch  diam.    '  490  " 
104  small  bolts  for  bottom  chord  each  4  inches  long 

by  f  inch  diam.  326  " 

218  nuts  each  J  Ib.  109  « 

264  lineal  feet  bottom  chord  4  x  }  1,584  " 

7,243 
Wood. 

12  floor-beams    7  x  14  14  feet  long  1,372  ft.  B.  M. 

2  track-strings  8  X  10  66       "  880       " 

1800  feet  B.  M.  floor  plank  1,800      " 

Total  board  measure  4,052 


Approximate  Estimate  of  Cost. 

15,549  Ibs.  castings  @  2|  cts.  $388  72 

7,243  ,"    rolled  iron  @  4  cts.  289  72 

4,000  feet  B.  M.  lumber  @  $15  60  00 

Making  eyes  on  104  diagonal  ties  @  30  cts.  312  00 

"         10  lateral  bolts  @  25  cts.  2  50 

"       208  small      "     @  10    "  20  00 

Workmanship  in  fitting  and  raising  66  feet  @  $5  330  00 

$1,403  74 
Estimated  cost  per  foot  lineal  $21  75. 


Calculation. 

The  weight  of  the  bridge  as  determined  by 

the  bills  of  materials  is  35,000  Ibs. 

Weight  of  maximum  load  1  ton  per  foot  132,000  " 

Total  weight  167,000  " 

Weight  on  one-half  the  bridge  83,500  " 

The  pressure  on  the  upper  chord  is  192,500  " 


236  BRIDGE   CONSTRUCTION. 

And  is  equivalent  to  12,000  Ibs.  sq.  in. 

Tension  on  lower  chord  48,000         " 

The  ties  and  braces  form  three  distinct  systems.  The 
proportions  of  weight  sustained  by  each  may  not  be  equal, 
and  cannot  be  estimated  with  certainty,  as  one  system  may 
be  brought  into  a  higher  degree  of  tension  than  another  by 
driving  the  wedges  unequally.  In  making  a  calculation? 
however,  it  will  be  assumed  that  they  bear  equally  and  each 
one-third  of  the  weight. 

The  weight  at  the  end  being  83,500  Ibs.,  the  tension  in  the 
direction  of  the  diagonals  will  be  116,900  Ibs.,  or  38,966  Ibs.  to 
each  system.  This  is  resisted  by  two  ties,  the  united  cross- 
section  of  which  is  three  inches,  making  the  tension  12,988 
Ibs.  per  square  inch. 

The  result  of  this  calculation  shows,  that  with  the  dimen 
sions  assumed  the  ties  are  stronger  than  the  chords,  and  that 
heavier  proportions  are  required  to  sustain  a  load  of  one  tan 
per  foot  in  addition  to  the  weight  of  the  structure. 

For  lighter  loads  the  bridge  is  sufficient,  and  by  increasing 
the  dimensions,  the  trusses  can  be  made  as  strong  as  may  be 
necessary  for  ordinary  spans. 

When  the  top  chord  extends  above  the  roadway  so  that  it 
cannot  be  braced  laterally,  it  is  very  important  that  its  hori 
zontal  dimension  should  be  increased  as  much  as  possible  to 
prevent  lateral  flexure. 

The  dimensions  used  in  the  calculation  are  those  of  a 
bridge  at  109th  street  in  the  city  of  New-York,  as  reported  to 
the  writer ;  they  may  not  be  entirely  correct  in  every  particu 
lar.  The  calculation  has  been  made  for  a  bridge  of  two 
trusses,  for  the  sake  of  uniformity,  as  the  other  calculations 
bave  been  made  in  the  same  way. 


CUMBERLAND   VALLEY   RAILROAD    BRIDGE.  237 

4 
CUMBERLAND  VALLEY  RAILROAD  BRIDGE 

ACROSS  THE  RIVER  SUSQUEIIANNA,  AT  HARRISBURG. 

The  original  contract  price  for  the  erection  of  the  super- 
itructure  of  the  bridge  was.  $52,000 ;  but,  in  consequence  of 
various  accidents,  the  actual  cost  of  construction  was  increased 
to  $62,000. 

It  was  used,  from  the  time  of  its  completion  until  Dec.  4, 
1844,  for  locomotive  engines,  and  was  without  roof;  on  this 
day  a  fire  occurred,  which  destroyed  all  but  four  spans  on  the 
Harrisburg  side  of  the  river.  It  was  quickly  rebuilt  on  the 
same  general  plan  as  the  original  structure,  with  some  slight 
alterations  in  the  details ;  the  hand-rail  was  omitted  on  the  top, 
and  a  pointed  roof  substituted.  On  the  new  bridge  locomo 
tives  are  not  allowed  to  pass. 

As  at  present  constructed,  the  bridge  is  an  ordinary  double 
lattice,  the  spans  vary  in  length  from  170  to  180  feet.  There 
are  23  spans  in  all,  and  the  total  length  of  the  bridge  is  4,277 
feet,  making  the  average  186  feet  from  centre  to  centre  of 
pieces,  or  176  feet  in  the  clear.  The  bridge  is  graded  with 
one  inclination  towards  the  eastern  shore,  of  19  feet  10  inches 
in  length  of  the  bridge.  There  are  two  roadways  on  the  lower 
chords,  each  11  feet  1  inch  from  centre  to  centre  of  trusses,  or 
9  feet  1  inch  in  clear  of  chords. 

Between  the  carriage-ways  was  a  space  of  6J  feet,  designed 
for  the  accommodation  of  foot-passengers,  but  it  was  found 
necessary  to  use  this  space  for  diagonal  bracing.  A  single 
railroad  track  is  on  the  top  of  the  bridge,  supported  by  the 
middle  trusses,  which  are  double  lattice,  while  the  outside 
trusses  are  single. 

The  trenails,  or  lattice-pins,  are  of  oak,  1 J  inches  in  diame 
ter,  there  are  4  at  each  intersection  of  the  chords,  and  3  at  the 
intermediate  intersections. 

The  total  height  of  the  outside  trusses,  from  the  top  of  the 
upper  chord  to  the  bottom  of  lower  chord,  is  18J  feet.  From 


238  BRIDGE   CONSTRUCTION. 

the  middle  of  the  upper  chord  to  the  middle  of  the  second 
chord  is  2  feet  9  inches. 

Total  length  of  middle  trusses  15  feet  9  inches. 

Bill  of  Timber  for  One  Span  of  186  feet. 

13,392  lineal  feet  chord  plank  3  X  12  40,176  ft 
176  lattice  plank  for  out 
side  trusses  3  x     9  24  ft.  long    9,504  " 
352  lattice  plank  for  inside 

trusses  3  x     9  20       "       15,840  " 

88  lower  floor-beams  4  x  10  11       "         3,227  " 

22         "          foot-path  4  x  10  6       "           440  " 

44  upper  floor-beams  5  x     7  20       "         2,569  " 

372  lineal  feet  track  strings  6x8  1,488  " 
22  lower  cross-pieces  be- 


22 

88 
12 
12 
44 
44 

tween  middle  trusses   5 
upper  cross-pieces  be 
tween  middle  trusses    5 
knee-braces                      4 
diagonal  braces                5 
5 
roof  braces                      4 
4 

x 
x 

X 

X 

x 

X 
X 

7 

7 
5 
6 
6 
3 
3 

8 

6 
5 

12* 

10 

7 
9 

it 

tt 
tt 
tt 
a 
tt 
a 

513 

385 
739 
375 
300 
308 
396 

a 

tt 
tt 
a 
tt 

a 
a 

88 

rafters 

4 

X 

5 

17 

" 

2 

,500 

tt 

10, 

000 

lineal  feet 

lath 

2* 

X 

1 

a 

2 

,083 

a 

88 

lower  lateral  braces        2 

X 

6 

10 

tt 

880 

tt 

88 

upper 

tt 

3 

X 

7 

101 

a 

1 

,617 

tt 

8, 

500 

feet  B.  M. 

21  inch 

oak  floor  plank 

8 

,500 

u 

5, 

500 

tt        tt 

1      " 

boards 

for 

upper  floor 

5,500 

it 

5, 

500 

tt        tt 

1      « 

tt 

sides 

5 

,500 

" 

2 

wall  plates 

5 

X 

12 

30ft. 

long 

300 

tt 

10 

bolsters 

7 

X 

9 

15 

tt 

789 

tt 

2 

pier  pieces 

4 

X 

10 

24 

tt 

160 

tt 

104,089 
Also  26,000  shingles 

1,760  pins,  18  inches  long,  1J  diameter 
1,056     "     30  «          li         " 

1,320     «       6          "          li        " 


CUMBERLAND   VALLEY   RAILROAD   BRIDGE.  239 


Iron  Rods. 

12  brace  rods  1  in.  13 J  ft.  long  162  ft.'] 

12         «  1  "     7J       "         87  "    ! 

12  upper  floor  rods  1  "    20         "       240  "    j>  736  lineal  ft, 

12      «  "  1  "    13J-       "       160  " 

12      "  "  1  "     7J-      "        87  " 

Total  weight  of  iron  1,950  pounds. 

"  timber  (one  span)  312,267       " 

"  shingles  36,720       " 


Total  weight  of  one  span     350,937       " 
Weight  per  lineal  foot         1,900       " 


Estimate  of  Cost  of  One  Span. 

104,089  feet  B.  M.  timber  @  $10J  $1092  93 

26,000  shingles  @    10J  273  00 

1,950  pounds  iron  rods  @  4  cts.  78  00 

Work  on  60  rods  and  nuts  @          57    "  30  00 

Workmanship  on  186  lineal  feet     @      6  58    "  1224  00 

§2697  93 

Average  cost  per  lineal  foot  $14  50. 
Cost  per  lineal  foot  of  single  track  bridge  $8  00. 


Data  for  Calculation. 

As  the  middle  trusses  sustain  the  weight  of  the  railroad 
track,  and  also  one-half  of  each  of  the  carriage-ways,  they 
will  bear  a  greater  proportion  of  the  load  than  the  trusses  on 
the  outside,  and,  therefore,  the  calculation  will  be  made  for 
them.     The  data  for  calculation  in  this  case  will  be, 
Span  between  supports  176  feet. 
Cross-section  of  upper  chords,  each  216  inches. 
"  lower       "          "     108      " 


240  BRIDGE   CONSTRUCTION. 

From  centre  to  centre  of  top  and  bottom  chords  14  ft.  9  in 
"         "  "  middle  "         9  "   3  " 

Number  of  intersections  in  one  span  between  supports,  42. 

Proportion  of  weight  of  bridge  on  middle  trusses,  175,000 
pounds. 

Greatest  accidental  load  from  a  train  of  cars  and  two 
loaded  wagons  in  middle  of  span,  200,000  pounds. 

Total  weight  on  one  span — two  trusses,  375,000  pounds. 

Distance  of  centre  of  gravity  from  end,  45  feet. 

The  spans  being  framed  continuously,  the  resistance  of  the 
chords  at  the  piers  may  with  propriety  be  taken  into  considera 
tion,  and  the  effect  will  be  equivalent  to  doubling  the  resist 
ing  areas  of  the  chords  in  the  centre. 

It  will,  therefore,  be  assumed  that  the  resisting  area  of  each 
of  the  upper  and  lower  chords  will  be  648  square  inches. 

The  neutral  axis  in  this  case  will  be  in  the  centre  of  the 
trusses.  If  the  strain  per  square  inch  at  the  extreme  chords 
be  represented  by  P,  the  strain  on  the  middle  chords  will  be  P  x 

4-6 

=—7 :  and  the  equation  of  equilibrium  will  be  648  P  x  7*4  y 

2  +  403  P  x  4-6x2  =  187,500  x  45;  or  13,297  P  = 
8,437,500 ;  or  P  x  635  pounds  =  strain  per  square  inch  upon 
the  chords. 


Strain  upon  the  Ties. 

The  strain  upon  the  diagonal  ties  and  braces  is  very  diffi 
cult  to  estimate  correctly.  The  following  considerations  will, 
perhaps,  lead  to  nearly  correct  conclusions  in  reference  to  the 
principle  upon  which  a  calculation  may  be  attempted. 

1.  Whatever  may  be  the  particular    arrangement    of  the 
parts,  if  the  weight  is  uniformly  distributed,  there  must  be  a 
gradual  increase  of  vertical  pressure  from  the  middle  to  the 
ends. 

2.  By  reference  to  the  plate  it  will  be  perceived  that  there 
are  six  separate  and  independent  systems  of  ties  and  braces, 
each  similar  to  that  exhibited  in  the  annexed  figure. 


CUMBERLAND  VALLEY  RAILROAD   BRIDGE.  241 


3.  These  systems,  even  if  the  workmanship  be  supposed 
to  be  perfect,  cannot  assist  equally  in  sustaining  the  load,  but 
the  portion  sustained  by  each  will  be  nearly  as  the  weights 
upon  the  points  5,  £,  d,  e,f,  and  g.  In  the  present  case,  there 
are  21  spaces  between  A  and  the  middle  of  the  span,  and  the 

15 

weight  at  g  will  be  ^  of  the  weight  at  A,  and  as  the  whole 

weight  must  be  sustained  by  the  systems  which  terminate  be 
tween  A  and  g,  the  portion  upon  one  system  at  A  must  be  21 

21 

~  (16  +  IT  +  18  +  19  +  20  +  21)  =  -r-  of  the   weight  of 

one-half  the  span,  or  nearly  one-fifth  the  weight. 

The  strain  upon  the  diagonal  is  14  times  the  vertical  pres 
sure,  therefore,  -?  X  14  =  ^  =  greatest  proportion  of  weight 

sustained  by  any  one  system. 

As  the  weight  of  the  half  span  loaded,  is  187,500  pounds, 

14 

we  will  have  187,500  X  ^  =  52,500  pounds. 

As  there  are  four  truss  frames,  each  will  bear  one-fourth,  and 
•  -  j—     =  13,125  pounds  is  the  greatest  force  either  of  tension 

or  compression  that  any  single  lattice  plank  will  be  required  to 
sustain. 

The  resisting  cross-section  of  each  plank,  after  deducting 
pin-holes,  is  15  square  inches,  and  the  strain  per  square  inch 
will  consequently  be  875  pounds. 

The  lattice  plank  in  the  direction  of  one  of  the  sets  of 
diagonals  being  in  a  state  of  compression,  and  the  others  in  a 
state  of  tension,  the  effect  upon  the  truss  is  to  produce  torsion. 
And  it  is  generally  observed  that  ordinary  lattice  bridges  yield 
by  twisting  or  warping  before  they  fail  in  any  other  way. 
-16 


242  BRIDGE   CONSTRUCTION. 


TRENTON  BRIDGE.     (Plate  9.) 

This  bridge  was  built  across  the  Delaware  River  at  Tren 
ton  in  the  year  1804,  by  Lewis  Wernwag.  It  is  supported  by 
five  trusses,  leaving  four  intervals  for  two  carriage-ways  and 
two  footpaths.  Each  truss  consists  of  a  single  arch  composed 
of  eight  planks  4  x  12  placed  in  contact  with  each  other. 
The  roadway  is  suspended  by  chains  of  1J-  inch  square  iron, 
the  links  of  which  are  about  4  feet  long,  and  5  inches  wide, 
passing  flatways  through  the  arches  and  between  the  chords 
and  counter-braces ;  a  key  passing  through  the  link  on  the  top 
of  the  arch. 

The  counter-braces  are  in  pairs  6  X  10,  spiked  to  the  chords 
at  the  lower  ends,  and  connected  with  the  arch  at  the  upper 
end  by  means  of  iron  straps  2  x  J  inch. 

The  chords  are  also  in  pairs  6 J  X  13J,  placed  in  contact ; 
between  them  the  links  of  the  suspension-chains  pass. 

The  floor-beams  are  suspended  below  the  roadway  under 
the  cnords,  and  held  in  place  by  the  suspension-chains,  the 
lower  links  of  which  pass  around  them. 

The  chords  are  connected  with  the  arches  at  the  end,  by 
means  of  long  straps  of  iron  passing  around  the  end  of  the 
arch  at  the  skew-back,  and  bolted  through  the  chords.  The 
width  of  each  carriage-way  in  the  clear  is  11  feet,  and  of  each 
footpath  6  feet. 

On  the  sides  are  large  spur  arches,  of  the  same  dimensions 
as  the  main  arch  of  the  truss,  extending  from  a  point  8  feet 
outside  of  the  truss  on  the  abutments  and  piers,  and  terminat 
ing  within  44  feet  of  the  centre— spiked  at  the  point  of  inter 
section  to  the  arches  of  the  main  truss.  During  the  present 
year  changes  have  been  made  by  removing  the  outside  truss 
on  the  lower  side,  to  a  sufficient  distance  to  convert  the  foot 
path  into  a  carriage-way.  Cast-iron  shoes  were  also  placed 
under  the  ends  of  the  counter-braces. 


DESCRIPTION   OF   AN   IRON   ARCHED   BRIDGE.          243 

The  bridge  is  without  lateral  braces ;  its  width  appears  to 
be  sufficient  to  prevent  lateral  motion.  The  spur  arches  also 
Assist  in  resisting  the  force  of  the  wind. 


DESCRIPTION  OF  AN  IRON  ARCHED  BRIDGE  OF 
133  FEET  SPAN, 

ACROSS  THE  CANAL  ON  SECTION    FIVE    OF    THE   PENNSYLVANIA  CEN 
TRAL   RAILROAD. 

The  chief  peculiarity  of  this  bridge  consists  in  its  iron 
arch,  which  is  extended  to  a  very  considerable  span,  and  fur 
nishes  a  highly  important  practical  test  of  the  powers  of  re 
sistance,  both  of  the  material  itself  and  of  the  particular  form 
in  which  it  is  employed.  At  the  same  time,  the  application  of 
the  principle  upon  which  the  structure  is  erected  has  been 
made  under  circumstances  which  render  it  perfectly  safe ;  for 
in  the  event  of  the  failure  of  the  arch,  the  truss,  without  it,  is 
more  than  sufficient  to  sustain  the  greatest  load  that  can 
come  upon  the  bridge. 

The  general  arrangement  of  the  truss  is  that  of  a  Howe 
bridge,  consisting  of  top  and  bottom  chords  of  wood,  with 
braces,  counter-braces,  and  vertical  rods.  The  braces  are  in 
pairs,  and  the  arches  pass  between  them.  The  counter-braces 
rest  upon  the  arches,  and  are  adjusted  by  means  of  set  screws 
above  and  below. 

The  arch  is  constructed  of  a  centre  rib  of  cast-iron,  7 
inches  deep,  with  upper  and  lower  horizontal  flanches,  5 
inches  wide ;  two  rolled  iron  plates  are  placed  on  the  top,  and 
two  on  the  bottom  of  the  cast  rib,  breaking  joint  with  the  rib 
and  with  each  other,  and  secured  by  clamps  at  proper  inter 
vals.  Below  the  chords  are  solid  cast-iron  skew-backs ;  and 
castings,  of  suitable  form  to  connect  with  the  skew-back  and 
receive  the  ends  of  the  arch,  are  placed  on  the  top  of  the 
ower  chord. 

Believing  that  the  failure  of  cast-iron  bridges  results  gene- 


244  BRIDGE    CONSTRUCTION. 

rally  from  the  inequality  of  pressure  upon  the  joints,  it  was 
proposed  to  ohviate  this  difficulty  in  the  present  case,  by  inter 
posing  plates  of  annealed  copper  between  the  ends  of  the  seg 
ments,  so  that  if  the  arch  should  rise  or  fall  by  expansion  or 
contraction,  the  comparatively  yielding  quality  of  the  inter 
posed  material  would  distribute  the  pressure,  and  prevent  the 
fracture  which  might  be  produced  if  the  joint  should  open, 
and  the  pressure  be  thrown  upon  the  upper  or  lower  corners 
of  the  castings. 

This  intention  was  defeated  by  circumstances  which  ren 
dered  it  necessary  to  hasten  the  completion  of  the  work.  The 
ribs  were  raised  without  dressing  the  joints,  and  the  copper 
plates  were  therefore  rendered  useless,  the  inequalities  of  sur 
face  being  too  great  to  admit  of  their  being  advantageously 
employed.  Under  these  circumstances  a  substitute  was  used, 
which  gave  more  satisfaction  than  could  have  been  obtained 
by  an  adherence  to  the  original  design,  and  was  much  more 
economical.  The  joints  were  separated  to  the  distance  of 
one-fourth  of  an  inch,  and  filled  with  spelter  poured  into  them 
in  a  melted  state ;  this  was  very  conveniently  done  by  bind 
ing  a  piece  of  sheet-iron  around  each  joint,  and  covering  it 
with  clay.  The  material  introduced  being  nearly  as  hard  as 
the  iron  itself,  and  filling  all  the  inequalities  of  the  surface, 
rendered  the  connection  perfect. 

The  pieces  of  castings  were  made  with  inch  holes  near 
the  ends,  through  which  rods  were  passed  horizontally  to 
assist  in  raising  them.  To  support  them  when  raised  to  their 
proper  positions,  pieces  of  board  were  nailed  vertically  from 
the  top  to  the  bottom  chord,  on  each  side  of  the  truss,  and 
short  rods  were  passed  through  the  holes  in  the  ends  of  the 
castings,  and  through  augur  holes  in  the  boards.  By  this 
arrangement  the  segments  were  held  securely,  and  no  ob 
struction  was  offered  to  the  attachment  of  the  arch-plates, 
which  were  added  by  clamping  one  end,  and  springing  them 
around  the  arch  by  a  rope  attached  to  the  other. 

The  most  important  advantage  that  was  expected  to  be 
derived  from  the  peculiar  arrangement  exhibited  in  this  struc 
ture,  was  a  practical  test  of  the  power  of  resistance  of  a 


DESCRIPTION   OF   AN  IRON   ARCHED   BRIDGE.          245 

counter-braced  iron  arch  on  a  large  scale,  under  circumstances 
which  would  render  its  failure,  under  any  possible  contin 
gency,  unattended  with  risk. 

It  has  been  observed  that  counter-braces  are  placed  above 
the  arch,  resting  against  it  by  means  of  adjusting  or  set 
screws.  In  addition  to  this,  there  is  a  vertical  post  of  oak 
between  each  pair  of  suspension  rods,  also  terminating  in  a 
set  screw  resting  on  the  arch.  It  will  be  readily  perceived 
that,  by  loosening  the  lower  counter-brace  screws,  and  by 
tightening  those  on  the  posts,  the  bridge  will  be  raised  upon 
the  arch,  and  the  latter  will  then  bear  the  whole  weight  both 
of  the  truss  and  its  load.  If  the  arch  should  prove  unable  to 
sustain  this  pressure,  the  truss  would  sink  again  to  its  original 
position  and  receive  the  weight. 

The  experiment  thus  far  has  been  entirely  successful,  and 
shows  that  the  counter-braced  arch,  wrhich  is  the  lightest  and 
cheapest  system  possible,  is  perfectly  reliable  for  spans  of  any 
magnitude ;  it  is,  in  fact,  a  satisfactory  test  both  of  the  princi 
ple  and  of  the  material. 

The  manner  of  adjusting  the  trusses,  and  the  observations 
subsequently  made,  were  as  follows : — 

1.  All  the  lower  counter-braces  were  unscrewed. 

2.  All  the  upper  counter-braces  were    tightened,  but   not 
screwed  hard. 

3.  The  levels  were  taken  from   a   permanent   level   mark 
at  the  foot  of  every  suspension  rod. 

4.  The  set  screws  upon  the  posts  were  tightened   by  two 
men,  with   suitable  wrenches,  beginning   at   the  middle  and 
proceeding  towards  the  ends ;  after  once  going  over,  the  level 
was  again  taken,  and  the  bridge  found  to  be  raised  one-fourth 
of  an  inch.     The  same  operation  wras  repeated,  and  the  rise 
found  to  be  half  an  inch,  which  was  sufficient  to  make  it  cer 
tain  that  the  whole  weight  was  upon  the  arch. 

5.  The  upper  counter-braces  were  examined  and  found  to 
have  become   loosened ;    they  were   again   screwed   up ;    the 
main-braces  were  loose ;  —  all  of  which  were  necessary  conse 
quences  of  raising  the  truss    upon    the    arch.     The  time  at 
which  these  adjustments  were  made  was  about  11  o'clock,  in 
July ;  the  weather  warm. 


246  BRIDGE   CONSTRUCTION. 

6.  The  bridge  was  again  examined  on  the  following  morn* 
ing ;  the  weather  was  cool,  and  the  contraction  of  the  arch, 
from  difference    of  temperature,  had  caused   the   posts   and 
counter-braces  to  become  loose,  whilst  the  main-braces  were 
found  to  be  in  full  action ;  in  other  words,  by  the  contraction 
of  the  arch  the  weight  was  again  thrown  upon  the  truss. 

7.  The   post    and   upper  counter-brace   screws  were  once 
more  tightened,  and  as  the  heat  of  the  day  increased,  the  arch 
expanded  and  lifted  the  bridge  to  a  greater  height  than  on  the 
preceding  day. 

8.  While  in  this  condition,  the  whole  weight  being  upon 
the  arch,  the  posts  and  upper  counter-brace  screws  tight,  and 
the  lower  counter-brace  screws  loose,  a  locomotive  was  passed 
over  the  bridge,  and  observations  made  with  a  level  and  rod 
whilst    it  was    running    repeatedly  backwards   and  forwards. 
The  greatest  variation  from  the  level  of  repose  was  J-  inch. 
The    arch    rose    slightly  when  the  locomotive  was  upon  the 
opposite  side,  and   fell   as  much   below  its  original   position 
when  it  was  on  the  same  side  as  that  upon  which  the  obser 
vation  was  made.      This  was  the   effect  anticipated ;   it  was 
not  to  be  supposed  that  the  arch,  in  the  condition  it  then  was, 
would  be  perfectly  rigid,  as  the  lower  counter-braces  were  all 
unscrewed,  and  the  upper  ones,  as  their  resistance  was  not 
transmitted  to  the  opposite  extremities  of  the  diagonals  of  the 
panels,  could  not  act  with  full  effect. 

9.  After  an  interval  of  several  days,  during  which  the  arch 
and  truss  experienced  no  change,  except  that  the  lower  coun 
ter-braces  were  screwed  up,  a  23-ton  locomotive  was  passed 
several  times  over  the  bridge.     No   level  was  at  hand  with 
which  to  make  observations  instrumentally,  but  the  eye  could 
detect  no  motion  in  the  arch;  it  appeared  to  be  perfectly  rigid 
in  every  direction,  and  a  subsequent  careful  examination  after 
more  than  one  year  of  service  cannot  detect  the  slightest  open 
ing  or  compression  at  the  joints. 

The  observations  made  thus  far  have  been  sufficient  to 
satisfy  the  writer  of  the  correctness  of  his  views  in  regard  to 
the  strength  and  rigidity  of  a  counter-braced  arch,  and  its  ap 
plicability  to  spans  of  great  extent.  Upon  this  principle,  an 


IRON   BRIDGE   OVER   RACOON   CREEK.  247 

iron  bridge  can  be  constructed  at  less  expense  than  is  now 
sometimes  incurred  in  the  erection  of  wooden  ones,  and  the 
durability,  with  proper  care,  is  almost  unlimited, 

An  iron  arch,  constructed  in  a  manner  similar  to  the  above, 
would  be  perhaps  the  cheapest  and  best  support  for  an  aque 
duct.  As  the  load  in  this  case  is  always  nearly  constant  and 
uniform,  the  curve  of  the  arch  should  be  a  parabola. 

No  practical  difficulty  need  result  from  expansion  and  con 
traction,  particularly  if  iron  tie-rods  are  not  used  for  the  lower 
chords.  The  counter-brace  rods  can  be  so  proportioned  and 
disposed  as  to  compensate  for  changes  in  the  arch,  and  keep 
the  tension  constant. 


IRON  BRIDGE  OVER  RACOON  CREEK, 

PENNSYLVANIA  RAILROAD.      (Plate  11.) 

This  bridge  depends  for  its  support  upon  4  counter-braced 
arches,  constituting  a  single  system,  unconnected  with  any 
self-supporting  truss.  The  arches  are  in  pairs,  one  on  each 
side  of  each  truss ;  they  are  composed  of  plates  of  malleable 
iron,  1  inch  by  3  inches,  placed  one  upon  another — 3  at  top 
and  3  at  the  bottom  of  each,  separated  by  pedestals  and  diag 
onal  braces,  and  secured  in  place  by  wrought-iron  clamps  and 
bolts.  There  are,  consequently,  in  the  two  trusses  24  leaves 
or  plates,  1  by  3,  arranged  in  groups  of  3,  the  separate  plates 
breaking  joints  with  each  other. 

The  diagonal-braces  between  the  arches  are  connected  by 
iron  keys,  kept  in  place  by  the  clamp  bolts. 

Each  skew-back  has  4  box-shaped  cavities  to  receive  the 
ends  of  the  plates. 

The  top  chord  is  of  wood,  12  by  12  inches.  The  lateral, 
diagonal,  and  counter-braced  rods  pass  through  it,  and  are  se 
cured  by  cast-iron  angle-blocks,  and  nuts  on  the  outside. 

The  roadway  is  on  top.  The  weight  is  transmitted  to  the 
irch  by  means  of  hollow  columns  or  cylinders.  Each  cylin- 


248  BRIDGE   CONSTRUCTION. 

der  is  capped  with  a  circular  plate,  which  is  cast  with  projec 
tions  on  both  sides,  fitting  into  the  bottom  of  the  chord  and 
into  the  top  of  the  column.  The  lower  ends  of  the  columns 
rest  in  sockets.  The  socket-boxes  have  cylindrical  projections, 
3  inches  in  diameter,  upon  the  sides,  which  fit  into  openings  in 
the  pedestals,  and  the  pedestals  rest  between  the  arches,  being 
firmly  held  in  place  by  the  clamps  and  diagonal  braces.  The 
cylindrical  form  of  the  socket-box  admits  of  a  vertical  position 
for  the  post  at  every  point. 

This  description  cannot  readily  be  understood  except  by 
reference  to  the  plates. 

The  small  posts  or  columns  which  connect  the  system  of 
counter-braces  pass  entirely  through  the  socket-boxes  and  out 
side  cylinders,  and  are  of  uniform  length,  extending  from  the 
top  to  the  bottom  chord.  They  are  connected  with  the  bottom 
chord  by  the  ends  of  the  diagonal  rods,  which  pass  through 
them  and  serve  as  bolts. 

The  action  of  the  system  is  the  reverse  of  that  which  takes 
place  in  an  ordinary  truss.  There  is  no  tension  on  the  lower 
chord  in  the  middle  ;  this  may  be  disconnected  without  injury. 
A  strain  upon  the  counter-brace  rods  produces  a  tension  on  the 
lower  chord  at  the  skew-back,  with  which  it  is  securely  con 
nected,  but  none  in  the  middle  of  the  span,  where  there  is  a 
coupling  link  to  allow  of  expansion  and  contraction. 

The  lower  lateral  bracing  is  by  means  of  diagonal  rods  and 
cross-beams  of  iron ;  the  upper  bracing  consists  of  wooden 
braces  and  lateral  rods  perpendicular  to  the  chords. 

Bill  of  Materials. 

CAST-IRON. 

4  skew-backs  592  pounds 

28  pedestals  714      « 

32  right  diagonal  arch  braces    )  Q  79 

32  left          "  "  / 

2  exterior  cylinders  3T35  inches  long,  diame 
ter  3J  and  5f  inches 

Amount  carried  up         4,048       " 


IRON   BRIDGE   OVER   RACOON   CREEK.  249 

Amount  brought  up  4,048  pounds, 
4  exterior  cylinders  6  j  inches  long,  diame 
ter  3f  and  5f  inches  96  " 
4  exterior  cylinders  18T]g  inches  long,  dia 
meter  3}  and  5£  inches  264  " 
4  exterior  cylinders  36]-jf  inches  long,  dia 
meter  3}  and  5f  inches  556  " 
14  interior  cylinders,  5  ft.  6{J  in.  long,  dia 
meter  3}  inches  1,596  " 
14  cap-plates  98  " 
14  socket-boxes  476  " 
7  girders  805  « 
14  lateral-brace  blocks  126  " 
34  small  angle  blocks  69  " 
34  keys  for  arch-braces  85  " 

Total  weight  of  castings  8,219  " 


MALLEABLE    IRON. 

62  plates  for  arches,  3x1,  various  lengths  to 

break  joint  11,559  pounds. 

86  coupling  plates,  f  X  2},  16  inches  long  655 

192  £  in.  bolts  with  head  and  screw,  14 J  in. 

long  358 

192  nuts  2J  x  f  305 

4  skew-back  bolts  bent  at  an  angle  of  45°  at 
one  end,  and  furnished  with  nut  and 
screw  at  the  other  end,  the  bent  end 
having  an  eye  to  receive  the  end  of 
the  lateral  brace  rod,  17  inches  long  63 
4  coupling  links  for  lower  chords  33 

4  rods  for  horizontal  bracing,  1st  panels  3 
inches  at  one  end,  bent  at  an  angle  of 
90°  to  pass  through  the  skew-back 
bolt ;  nuts  and  screws  on  both  ends, 
length  from  angle  7  ft.  11J  inches. 

Amount  carried  over         1,314 


250  BRIDGE   CONSTRUCTION. 

Amount  brought  over     1,314  pound?.. 
4  lateral-brace  rods  for  2d  panels,  8  feet  11 

inches. 

4  lateral-brace  rods  for  3d  panels,  9  feet  long. 
4  "  4th    "       9  feet  |-  in. 

long. 
14  diagonal-brace  rods  with  nut   and   screw 

at  upper  end,  lower  er,d  bent  at  an 

angle  of  49°  from  straight  direction, 

to  pass  through  the  column  and  tie, 

secured  by  nut  and  screw  on  outside, 

length  of  whole  rod  9  feet  11  inches, 

elbow  7  inches. 
4  counter-brace  rods  for  1st  panels,  the  up 

per  end  passes   through   angle-block 

placed  on  top  of  chord  with  nut  and 

screw ;  the  lower  end  is  formed  into 

an    eye   to    embrace    the    skew-back 

bolt,  length  from  centre  of  eye,  8  feet 

11  inches. 
4  counter-brace   rods  for  2d  panels,  upper 

end  as  before ;  lower  end  bent  138° 

to  enter  the  bottom  of  the  cast-iron 

column,  the  bent  end  has  an  eye  2 

inches  long,  1J-  inch  wide,  the  cen 
tre  of  which  is  2J  inches  from  upper 

side  of  angle  of  rod,  length  of  rod 

from  angular  point  to  end  of  screw 

9  feet  4  inches.     Total  length  9  feet 

8  inches. 
4  counter-brace  rods  for  3d  panels  similar  to 

those  in  2d  panels,  length  9  feet  5 

inches. 
4  counter-brace  rods  for  4th  or  middle  pan 

els  as  above,  length  9  feet  5J  inches. 
7  rods  for  upper  lateral  braces,  7  feet  3  inches. 

Amount  carried  up         1,314       ** 


IROX   BRIDGE   OVER   RACOON   CREEK.  251 

Amount  brought  up  1,314  pounds. 
Total  length  of  the  53  inch  rods,  483  feet, 

weight  1,280        " 
4  lower  chords  48  feet  long,  1J  inches  dia 
meter,  705        " 
100  nuts  for  inch  bolts  1  X  2J  X  2J  141        « 

Total  weight  of  malleable  iron,  exclusive  of 

arch  plates  3,440        « 

TIMBER. 


2  upper  caps  12  x  12,  50  ft.  long                     600  feet  B.  M. 

16  lateral-braces  4  x  5,  8J      "                           126  «      " 

726  «     " 

Estimate. 

8,219  pounds  castings  at  2  cents  $164  38 

11,559       «       arch  plates  at  $57  per  ton  gross  294  11 

3,440       "      bolt  and  plate  iron  at  3J  cents  120  50 

726  feet  B.  M.  timber  at  $1%  per  M.  8  71 

Total  cost  of  materials  $587  70 

^  Workmanship . 

69  days'  work  making  plates  and  fitting  pieces 

at  $1  50  $103  50 

Files  3  50 

Making  53  inch  bolts  at  53  cents  13  25 

"       192  |  inch  bolts  at  10  cents  19  20 

Freight  and  tolls  for  delivery  of  materials  21  00 

4  men  6  days  raising  at  $1  25  30  00 

Total  cost  of  workmanship  190  45 

Cost  of  materials  per  foot  of  span  12  50 

"      work                 «  6  05 

Total  cost  per  foot  §18  55 


252  BRIDGE   CONSTRUCTION. 


Data  for  Calculation. 

Span  47  feet. 

Rise  of  arch  5     " 

Cross-section  of  all  the  arches  in  square  inches,  ex 
clusive  of  castings  72     " 
Distance  of  centre  of  gravity  from  abutment                  12     " 
Whole  weight  cf  bridge                                        27,500  pounds. 
Weight  of  bridge  and  load                                 122,000       " 

As  the  arch  sustains  the  whole  of  the  weight,  the  calcula 
tion  is  extremely  simple. 

Let  P  =  pressure  per  square  inch  at  crown. 

122000 
Then  P  x  5  x  72  =  — g —  +  12,  or  P  =  2033  pounds  per 

square  inch,  or  only  about  one-thirtieth  of  the  crushing  force. 

The  greatest  pressure  upon  any  one  post  may  be  taken  at 
6  tons.  The  cross-section  is  15  square  inches.  Pressure  per 
square  inch  800  pounds. 

The  projections  of  socket-boxes  are  3  inches  cylinder,  the 
pressure  on  each  is  3  tons  =  per  square  inch  630  pounds. 

It  is  unnecessary  to  calculate  the  strain  upon  the  counter- 
brace  rods,  they  are  evidently  sufficient ;  and  for  the  manner 
of  making  the  calculations,  sufficient  illustrations  have  already 
been  given.  As  a  general  rule  in  regard  to  counter-braces,  it 
may  be  stated,  that  their  dimensions  may  be  assumed  as  con 
stant,  whatever  may  be  the  span ;  or  rather,  the  counter-braces 
should  bear  a  fixed  proportion  to  the  width  of  the  panels, 
without  reference  to  any  of  the  other  dimensions  of  the  bridge, 
and,  consequently,  the  counter-brace  rods  need  not  be  larger  or 
more  numerous  in  proportion  to  the  length  in  a  bridge  of  large 
span  than  a  shorter  one. 

The  truth  of  this  assertion  will  be  evident  from  these  con 
siderations  : — 

The  greatest  possible  strain  upon  any  counter-brace  has  been 
shown  to  be  less  than  the  variable  load  upon  one  panel.  The 
weight  of  the  structure  produces  no  strain  whatever  upon  the 
counter-braces.  The  greatest  variable  load  on  railroad  bridges 


BALTIMORE   AND   OHIO    RAILROAD   BRIDGE.  253 


has  been  assumed  as  one  ton  per  foot  lineal,  which  will  be  one 
thousand  pounds  per  foot  lineal  for  each  truss.  If  the  panels 
are  10  feet  (which  is  nearly  an  average  for  the  bridges  on  the 
Pennsylvania  Railroad),  the  greatest  strain  upon  any  single 
counter-brace  will  be  10,000  pounds,  and  this  will  be  resisted 
by  a  single  square  inch  of  metal. 

As  a  general  rule,  which  will  save  much  trouble  in  calcula 
tion,  the  proper  cross-section  of  the  counter-brace  rods  for 
railroad  bridges  of  any  span  may  be  estimated  at  one  square 
inch  for  every  10  feet  of  truss. 


BALTIMORE  AND  OHIO  R.  R.  BRIDGE.     (Plate  12.) 

The  plan  of  this  bridge  was  furnished  by  B.  H.  Latrobe, 
Esq.,  Chief  Engineer.  It  is  an  admirable  combination,  pos 
sessing  every  essential  of  a  well-proportioned  and  scientifically 
arranged  structure.  It  is  a  system  of  counter-braces  and 
braces.  In  its  general  principle  it  bears  some  resemblance  to 
the  celebrated  bridge  across  the  Rhine  at  Schauffhausen,  but 
the  latter,  owing  to  the  absence  of  counter-braces,  was  so  flexi 
ble  that  it  would  vibrate  with  the  weight  of  a  single  man, 
whilst  the  Baltimore  and  Ohio  R.  R.  Bridge  is  so  rigid  that 
the  heaviest  locomotives,  running  with  great  velocity,  produce 
but  very  little  effect. 

These  bridges  possess  great  strength,  but  they  are  not  as 
economical  in  first  cost  as  many  others. 

The  calculation  for  the  strains  is  more  simple  than  in  any 
other  form  of  bridge  ;  each  set  of  arch-braces  is  to  be  considered 
as  sustaining  one-half  the  weight  of  the  interval  on  each  side 
of  it,  between  it  and  the  next  set  of  braces. 

Description  of  Details. 

Fig.  1  shows  the  manner  of  adjusting  the  horizontal  diago 
nal  brace,  in  tie-beams. 


254  BRIDGE   CONSTRUCTION. 

Fig.  2  shows  the  manner  of  adjusting  the  horizontal  braces 
in  floor-beams. 

Fig.  3  represents  the  skew-back,  which  is  cast  in  two 
pieces,  the  hindmost  part  a  a  being  the  buttress  of  the  main 
part  b  b.  The  heels  of  the  arch-braces  rest  in  cast-iron  shoes, 
between  which  and  the  abutting  steps  of  the  skew-back,  ad 
justing-screws  operate  to  push  the  braces  forward  when  re 
quired  in  raising  or  adjusting  the  truss. 

Fig.  4  shows  the  arrangement  of  the  chord-splices. 

Fig.  5  shows  the  intersection  of  the  counter-brace  and 
main-braces.  The  counter-braces  are  cut  off  at  their  intersec 
tions  with  the  main  or  panel-braces,  aud  their  connection  car 
ried  around  the  latter  by  means  of  the  cast  plates  shown  at 
d  d,  these  plates  are  connected  across  the  truss  by  bolts  pass 
ing  within  cast  tubes  acting  as  struts,  as  at//. 

Fig.  6  shows  the  connection  of  the  upper  tie-beams  with 
the  chords,  braces,  and  counter-braces. 

The  principal  rafters  foot  upon  the  casting  o  between  the 
tie-beams. 

Fig.  7  shows  the  manner  of  securing  the  rail  to  the  rail-joist 
or  string-piece.  The  rail-joists  and  floor-beams  are  tied  to 
gether  by  a  vertical  bolt  at  each  intersection.  The  rail  is 
fastened  to  the  rail-joist  by  double-headed  bolts. 

All  the  principal  abutting  surfaces  of  the  timbers  are  sepa 
rated  by  cast-iron  plates,  and  every  joist  has  an  independent 
adjustment  by  means  of  screw-bolts  or  wedges. 

Material  in  one  span  of  133  feet  in  the  clear  of  abutments, 
or  145  feet  from  end  to  end  of  skew-backs : 

Timber,  63,000  feet  B.  M. 

Cast-iron,  57,156  pounds. 

Wr  ought-iron,  15,340. 


CANAL  BRIDGE,  SECTION. 6,.  PENN.  RAILROAD. 

This  bridge  has  some  resemblance  to  that  on  Sec.  5,  ex 
hibited  in  Plate  10.     The  principal  differences  are  the  absence 


BOILER   PLATE    TUBULAR    BRIDGE.  255 

of  posts,  and  the  use  of  a  wooden  arch  composed  of  layers  of 
plank,  instead  of  an  arch  of  cast-iron. 

The  advantage  of  such  an  arrangement  is  the  great  facility 
which  it  affords  for  adjustment.  To  raise  the  camber  of  this 
bridge  it  is  not  necessary  to  remove  a  single  stick  of  timber ; 
all  that  is  required  is  to  slacken  the  counter-braces  and  tighten 
the  vertical  rods,  until  the  bridge  is  raised  to  a  sufficient  ex 
tent,  after  which  the  counter-braces  should  be  tightened.  Two 
men  in  less  than  an  hour  can  adjust  a  bridge  constructed  in 
this  way. 

The  arch  can  be  made  to  bear  any  proportion  of  the 
weight  by  tightening  the  counter-braces  on  the  upper  side. 


BOILER  PLATE  TUBULAR  BRIDGE.     (Plate  4.) 

(COPY   OP   A   LETTER   FROM   THE   INVENTOR.) 

Eeading,  May  1,  1849. 

DEAR  SIR  :  —  Inclosed  I  send  you  the  drawings  of  the 
three  bridges  I  constructed  on  the  Baltimore  and  Susquehanna 
Railroad  while  engaged  as  Superintendent  of  Machinery  and 
Road. 

The  one  marked  A  was  built  at  the  Bolton  depot  in  the 
winter  of  1846  and  '7,  and  was  put  in  its  place  in  April,  1847. 
This  bridge  is  made  of  puddled  boiler-iron  J  inch  in  thickness. 
The  sheets,  standing  vertical,  are  38  inches  wide  and  6  feet 
high,  and  riveted  together  with  f  rivets,  two  and  a  half  inches 
from  centre  to  centre  of  rivets.  You  will  observe  by  reference 
to  the  drawing,  that  each  truss-frame  is  composed  of  two  thick 
nesses  of  iron,  12  inches  distant  from  each  other,  and  con 
nected  together  by  T5G  iron  bolts,  passing  through  round  cast- 
iron  sockets  at  intervals  of  12  inches ;  which  arrangement,  to 
gether  with  the  lateral  bracing  between  the  two  trusses,  which 
is  composed  of  }  round  iron,  set  diagonally  and  bound  together 
at  the  crossing  by  two  cast-iron  plates  about  4  inches  diame 
ter,  the  sides  next  to  the  bracing  being  cut  in  such  a  man- 


256  BRIDGE    CONSTRUCTION. 

ner,  that  "when  the  two  f  bolts  that  pass  through  them  were 
screwed  up,  it  held  them  firmly  together.  There  is  also  a  bolt 
passing  through  both  truss-frames  and  through  the  heels  of 
the  lateral  bracing,  at  right  angles  with  the  bridge,  which  se 
cured  the  heels  of  the  lateral  braces,  and  by  means  of  a  socket 
in  the  centre  made  a  lateral  tie  to  the  bridge,  giving  the  bridge 
its  lateral  stability.  The  lower  chords  were  of  hammered 
iron,  there  being  some  difficulty  at  that  time  to  get  rolled  iron 
of  the  proper  size,  and  are  in  one  entire  piece,  being  welded 
together  from  bars  12  feet  long.  There  are  eight  of  them  5  X 
f  inches,  one  on  either  side  of  each  piece  of  boiler  iron,  and 
fastened  to  it  with  f  inch  iron  rivets  6  inches  distant  from 
each  other.  There  are  but  four  top  chords,  and  of  the  same 
size  of  the  bottom,  two  on  each  truss  near  the  top,  the  timber 
for  the  rail  making  up  the  deficiency  for  compression,  and  an- 
swering  the  purpose  of  chords.  This  bridge  was  built  at  the 
time  Messrs.  Stephenson  and  Brunell  were  making  their  ex 
periments  with  cylindrical  tubes  preparatory  to  constructing 
the  Menai  bridge ;  the  cylindrical  tubes  failing,  they  adopted 
this  plan  of  bridge.  The  entire  weight  of  the  bridge  is  14 
gross  tons,  and  cost  $2,200 ;  but  as  the  same  kind  of  iron  of 
which  the  bridge  is  composed  can  be  had  for  at  least  15  per 
cent,  less  now,  than  it  cost  at  that  time,  it  would  be  but  fair 
to  estimate  the  cost  of  the  bridge  at  $1,870,  without  any  refer 
ence  to  the  labor  that  is  misapplied  in  all  new  structures  of  the 
kind,  making  the  cost  of  a  bridge  55  feet  long  $34  per  foot. 
And  I  have  no  doubt,  where  there  would  be  a  large  quantity 
of  iron  required  for  such  purposes,  that  it  could  be  had  at  such 
prices  as  to  bring  down  the  cost  of  bridges  of  55  feet  length  to 
$30  per  foot. 

Very  respectfully  yours, 

JAMES  MILLHOLLAND 


ARCHED   TRUSS   BRIDGE.   READING   RAILROAD.         257 

ARCHED  TRUSS  BRIDGE,  READING  RAILROAD. 

(Plate  2.) 
(DIRECTIONS  GIVEN  BY  PATENTEE,  j.  D.  STEELE.) 

This  improvement  consists  in  combining  arches  with  a 
truss  frame  by  securing  them  to  tension  posts — "  a  a"  which 
posts  are  connected  to  the  chords  by  screw-fastenings,  "e  e," 
and  so  arranged  as  to  admit  of  changing  the  position  of  the 
arches  relatively  to  the  chords,  or  of  drawing  together  the 
chords  without  changing  the  position  of  the  arches,  by  which 
means  the  strain  can  be  regulated  and  distributed  over  the 
different  parts  of  the  bridge  at  pleasure. 

In  erecting  a  bridge  on  this  plan  it  will  be  found  desirable 
to  be  governed  by  the  following  directions,  viz. :  —  The  truss 
must  first  be  erected,  provided  with  suitable  cast-iron  skew- 
backs  to  receive  the  braces  and  tension  posts,  and  the  several 
parts  of  the  chords  should  be  connected  with  cast-iron  gibs. 
Wedging  under  the  counter-braces  must  be  avoided  by  extend 
ing  the  distance  between  the  top  skew-backs  sufficiently  to 
bring  the  tension  posts  on  the  radii  of  the  curve  of  cambre  of 
the  bridge.  The  tension  posts  must  be  about  eight  inches 
shorter  than  the  distance  between  the  chords,  and  in  screwing 
up  the  truss  care  must  be  taken  not  to  bring  their  ends  in  con 
tact  with  the  chords ;  but  they  must  be  equidistant,  and  about 
four  inches  from  them.  When  the  truss  is  thus  finished  it 
must  be  thrown  on  its  final  bearings,  and  it  is  then  ready  to 
receive  the  arches,  which  should  be  constructed  on  the  curve 
of  the  parabola,  with  the  ordinates  so  calculated  as  to  b& 
measured  along  the  central  line  of  the  tension  posts.  They 
must  be  firmly  fastened  to  the  posts  and  bottom  chords  by 
means  of  strong  screw-bolts  and  connecting  plates,  as  shown 
at  "cZ  £?,"  and  should  foot  on  the  masonry  some  distance  below 
the  truss,  which  can  be  done  with  safety,  as  the  attachment  to 
the  posts  and  chords  will  relieve  the  masonry  of  much  of  their 
horizontal  thrust.  When  a  bridge  so  constructed  is  put  int<r 
17 


•^58  BRIDGE    CONSTRUCTION. 

use  it  will  be  found,  as  the  timber  becomes  seasoned,  the  weight 
will  be  gradually  thrown  upon  the  arches,  which  will  ultimately 
bear  an  undue  portion  of  the  load.  To  avoid  this  the  cambre 
must  be  restored  and  the  posts  moved  up,  so  as  again  to  divide 
the  strain  between  the  truss  and  the  arches. 

This  adjustment  must  take  place  once  or  twice  in  each 
year,  until  the  timber  becomes  perfectly  seasoned,  after  which, 
in  a  well  constructed  bridge,  but  little  attention  will  be  re 
quired.  Plates  of  iron  should  in  all  cases  be  introduced  be 
tween  the  abutting  surfaces  of  the  top  chords  and  arches,  and 
all  possible  care  taken  to  prevent  two  pieces  of  timber  from 
coming  in  contact,  by  which  decay  is  hastened  ;  care  should 
also  be  taken  to  obtain  the  curve  of  the  parabola  for  the  arches, 
as  it  is  the  curve  of  equilibrium  and  of  greatest  strength,  as 
has  been  shown  by  experiment.* 

Bridges  constructed  on  this  plan  will  be  found  to  possess 
an  unusual  amount  of  strength,  for  the  quantity  of  material 
contained  in  them,  and  if  well  built  and  protected,  great  dura 
bility. 


BKIDGE.  ACROSS  THE  SUSQUEHANNA,  AT  CLARK'S 
FERRY.     (Plate  13.) 

This  bridge  has  been  selected  as  a  fair  specimen  of  the 
ordinary  Burr  bridge,  a  mode  of  construction  more  common 
than  any  other  in  Pennsylvania,  and  which  experience  for 
many  years  has  proved  to  be  one  of  the  best  arrangements  for 
ordinary  purposes. 

Probably  no  other  plan  has  ever  secured  more  general 
approbation  or  better  sustained  itself  than  the  Burr  bridge,  and 

*  The  parabola  is  the  curve  of  equilibrium  when  no  load  is  upon  the 
bridge,  and  also  when  the  load  is  uniform,  but  there  can  be  no  curve  of 
equilibrium  for  a  variable  load  of  a  passing  train.  Stiffness  can  1)0 
secured  in  this  case,  only  by  an  efficient  system  of  counter-braces. 

The  plan  proposed  fulfils  every  condition  of  a  good  bridge. —  Author 


SUSQUEIIANNA    BRIDGE   AT    CLARK'S   FERRY.         259 

it  is  certain  that  when  counter-braced  and  properly  propor 
tioned,  it  forms  a  truss  fully  sufficient  to  bear  the  heaviest 
railroad  train.  A  particular  description  is  unnecessary ;  the 
plates,  with  the  accompanying  bill  of  materials,  will  furnish 
the  engineer  or  builder  with  all  necessary  information,  and  a 
sufficient  number  of  examples  have  been  given,  to  show  how 
the  calculations  for  the  strains  are  to  be  made. 

These  bridges,  well  counter-braced,  have  been  recently 
introduced  into  New  England,  by  H.  R.  Campbell,  Esq.,  and 
are  remarkably  rigid  structures. 


Bill  of  Timber  for  One  Span. 

Length. 

Inches.        Feet.  Inches.  Feet. 

2  Wall  plates                             10    x  15  28  700 
30  Bottom  chords                          8     x  15  40  6  12,150 
15  Top                                        11    x  11  40  6  6,135 
21  Floor  beams  (large)                10     x  11  36  6,930 
21      «         «      (small)                 5    x  11  36  3,465 
36  Arch  pieces                              8    x  15  31  21,160 
48  Queen  posts                            11    x  14  17  6  10,752 

6      «         "                             11    x  14  23  6  1,812 

6      «         "                              11    x  14  22  1,692 

3  Ring        "                               11    x  18  17  6  867 
60  Main  braces                              8     x  11  14  6,180 

120  Check    «                                 4-J  x  11       96  4,800 

21  Tie  beams                                 9    x  10  28  4,410 

84  Knee  braces                              4x5       8  1,125 

80  Lower  laterals                          4     x    8  12  2,560 

40  Upper       "                               4    X    8  24  2,560 

120  Rafters                   3  X  5  and    3    x    6  14  6  2,640 

21  Roof  posts                               1J  x  11  36  100 

6,912  Feet  B.  M.  weather-boarding  3    x  12  13  6,912 

22,260  Shingles 

9,540  Lineal  feet  laths 

3  inch  flooring  plank  24  7,314 

Hand  railing                              2J  X     7  3,710 


2(30  BKIDGE   CONSTRUCTION. 


TOWING  PATH. 

Length. 

Inches.  Feet.  Inches.  Feet. 

7  Arch  pieces                                  6  x  15  32  1,344 

6  Lower  chords                                 6  X  12  40  6  1,458 

6  Upper      "       (railing  caps)         5  X  10  40  6  1,014 

1  Ring  post                                      9  x  12  8  72 
38  Queen  posts                                  9x9  8  2,042 

2  "         "                                    9x98  180 
2      "         "                                    9  x    9  12  6  168 

40  Braces                                            5x    9  7  1,200 

Weather-boarding,  square  feet  4,120 

Flooring  in  lengths  of  24  1,783 


IMPROVED  LATTICE  BRIDGES.     (Plate  14.) 

Amongst  the  great  variety  of  bridge  plans  that  have  from 
time  to  time  been  presented  to  the  American  public,  none  have 
experienced  a  more  favorable  reception  than  the  ordinary  lat 
tice.  Its  beautiful  simplicity,  light  appearance,  and  especially 
its  economy,  have  secured  for  it  the  favor  of  many  of  our  most 
eminent  engineers  and  builders,  as  is  evident  from  the  extent 
to  which  it  has  been  adopted  on  works  of  the  greatest  magni 
tude  and  importance. 

On  ordinary  roads,  and  on  railways  not  subjected  to  very 
heavy  transportation,  this  plan  of  superstructure,  when  well 
constructed,  has  been  found  to  possess  almost  every  desidera 
tum:  nevertheless,  experience  has  fully  proved,  that  unless 
strengthened  by  additional  arch-braces,  or  arches,  the  capacity 
of  the  structure  is  limited  to  light  loads,  and  spans  of  small 
extent.  The  public  works  of  Pennsylvania  furnish  abundant 
proof  of  the  truth  of  this  assertion  ;  and  several  railways  might 
be  enumerated,  on  which  the  lattice  bridges  have  from  neces 
sity  been  strengthened  by  props  from  the  ground,  by  arches. 
or  arch-braces,  added  when  the  insufficiency  of  the  structure 


IMPROVED    LATTICE   BRIDGES.  261 

was  found  to  require  it.  These  circumstances  have  produced 
a  change  in  opinion  hostile  to  the  whole  plan,  and  it  is  much 
to  be  regretted,  that  instead  of  introducing  such  modifications 
and  improvements  as  would  remedy  existing  defects  and  re 
tain  its  advantages,  other  plans  have  been  substituted  at  an 
expense  frequently  more  than  double  that  of  an  efficient  lattice 
structure. 

One  of  the  first  defects  apparent  in  some  old  lattice  bridges, 
is  the  warped  condition  of  the  side  trusses.  The  cause  which 
produces  this  effect  cannot  perhaps  be  more  simply  explained 
than  by  comparing  them  to  a  thin  and  deep  board  placed  edge 
ways  on  two  supports,  and  loaded  with  a  heavy  weight ;  so 
long  as  a  proper  lateral  support  is  furnished,  the  strength  may 
be  found  sufficient,  but  when  the  lateral  support  is  removed, 
the  board  twrists  and  falls. 

A  lattice  truss  is  composed  of  thin  plank,  and  its  construe 
tion  is  in  every  respect  such  as  to  render  this  illustration  ap 
propriate.  Torsion  is  the  direct  effect  of  the  action  of  an^ 
weight,  however  small,  upon  the  single  lattice. 

A  second  defect  may  be  found  in  the  inclined  position  of 
the  tie ;  all  bridge-trusses,  whatever  may  be  their  particular 
construction,  are  composed  of  three  series  of  timbers ;  those 
which  resist  and  transmit  the  vertical  forces  are  called  ties  and 
braces,  and  those  which  resist  the  horizontal  force  are  known 
by  the  names  of  chords,  caps,  £c. 

In  every  plan  except  the  common  lattice,  these  ties  are 
either  vertical,  or  perpendicular  to  the  lower  chords  or  arches, 
and  as  the  force  transmitted  by  any  brace  is  naturally  resolved 
into  two  components,  one  in  the  direction  of,  and  the  other  at 
right  angles,  to  the  chord  or  arch,  it  would  seem  that  this  latter 
force  could  be  best  resisted  by  a  tie  whose  direction  wras  also 
perpendicular.  The  short  ties  and  braces  at  the  extremities, 
furnishing  but  an  insecure  support,  render  these  points,  which 
require  the  greatest  strength,  weaker  than  all  others ;  this  de 
fect  is  generally  removed  by  extending  the  truss  over  the  edge 
of  the  abutment  a  distance  about  equal  to  its  height,  or  to  such 
a  distance  that  the  short  ties  will  not  be  required  to  sustain 
any  portion  of  the  weight,  the  effect  of  which  is  to  provide  a 


262  BRIDGE    CONSTRUCTION. 

remedy  at  the  expense  of  economy,  by  the  introduction  of  from 
fifteen  to  thirty  feet  of  additional  truss. 

A  bridge  whose  corresponding  timbers  in  all  its  parts  are 
of  the  same  size,  is  badly  proportioned ;  some  parts  must  be 
unnecessarily  strong,  or  others  too  weak,  and  a  useless  pro 
fusion  of  material  must  be  allowed,  or  the  structure  will  be 
insufficient. 

If,  for  example,  the  forces  acting  on  the  chords  increase 
constantly  from  the  ends  to  the  centre,  the  most  scientific 
mode  of  compensation  would  appear  -to  be,  to  increase  gra 
dually  the  thickness  of  the  chords ;  and  for  similar  reasons 
the  ties  and  braces  should  increase  in  an  inverse  order  from 
the  centre  to  the  ends. 

.In  accordance  with  this,  it  is  found  that  in  bridges  that 
have  settled  to  a  considerable  extent,  the  greatest  deflection  is 
always  near  the  abutment ;  that  is,  the  chords  are  bent  more 
at  this  point  than  in  the  centre,  and  the  joints  of  the  braces 
are  much. more  compressed.  It  is  also  found  that  the  weakest 
point  of  a  lattice  bridge  is  near  the  centre  of  the  lower  chord ; 
this  might  be  expected,  since  from  the  nature  of  the  force,  and 
the  mode  of  connection,  the  joints  of  the  lower  chords  are  only 
half  as  strong  as  the  corresponding  ones  of  the  upper  chord,  it 
being  assumed  that  the  resistances  to  compression  and  exten 
sion  are  equal.  This  defect  may  be  in  a  great  degree  removed 
by  inserting  wedges  behind  the  ends  of  the  lower  chords.  A 
variation  in  the  size  of  every  timber,  according  to  the  pressure 
it  is  to  sustain,  would  of  course  be  inconvenient  and  expen 
sive  ;  but  as  the  principle  of  proportioning  the  parts  to  the 
forces  acting  upon  them,  is  of  great  importance,  such  other 
arrangements  should  be  adopted  as  will  secure  its  advantages, 
and  at  the  same  time  possess  sufficient  simplicity  for  practice , 
this  is  effected  by  the  introduction  of  arch-braces  or  arches, 
than  which,  a  more  simple,  scientific,  and  efficacious  mode  of 
strengthening  a  bridge  could  not  perhaps  be  devised,  as  they 
not  only  serve,  with  the  addition  of  straining  beams,  to  relieve 
the  chords,  and  give  them  that  increase  of  thickness  at  the 
points  of  maximum  pressure,  which  is  essential  to  strength, 
but  they  also  relieve  the  ties  and  braces  by  transmitting  di- 


IMPROVED    LATTICE   BRIDGES.  263 

rectly  to  the  abutments,  or  other  fixed  supports,  a  great  part  of 
the  weight  that  they  would  otherwise  be  required  to  sustain. 

It  may  perhaps  be  objected,  that  the  pressure  of  the  arcli- 
braces  or  arches  would  injure  the  abutments  :  in  answer  to 
this,  it  may  be  remarked  that  a  certain  degree  of  pressure  is 
very  proper ;  the  embankment  behind  an  abutment  exerts  a 
very  great  force  upon  it,  the  tendency  of  which  is  to  push  it 
forward ;  if,  then,  a  counter-pressure  can  be  produced  by  the 
thrust  of  arch-braces,  or  by  wedging  behind  the  ends  of  the 
lower  chords,  two  important  advantages  are  gained ;  the  abut 
ment  is  not  only  increased  in  stability,  but  the  tension  on  the 
lower  chord  of  the  bridge  is  diminished  by  an  amount  equal  to 
the  degree  of  pressure  thus  produced. 

It  is,  however,  proper  to  observe,  that  when  the  situation 
of  the  embankment  exposes  it  to  the  danger  of  being  washed 
away  from  the  back  of  an  abutment,  the  pressure  on  its  face 
must  not  be  sufficient  to  destroy  its  equilibrium ;  should  this 
effect  be  apprehended,  the  horizontal  ties  must  be  sufficient  to 
sustain  the  thrust  of  the  bridge. 

An  essential  condition  in  every  good  bridge  is,  that  it  dliall 
not  only  be  sufficient  to  resist  the  greatest  dead  weight  that  it 
can  ever  be  required  to  sustain  in  the  ordinary  course  of  ser 
vice,  but  it  must  also  be  secure  against  the  effects  of  variable 
loads.  This  is  generally  effected  by  the  addition  of  counter- 
braces  ;  but  the  lattice  truss  possesses  this  peculiarity,  that  it  ia 
counter-braced  without  the  addition  of  pieces  designed  exclu 
sively  for  this  purpose :  to  prove  this,  invert  the  truss,  when  it 
will  be  apparent  that  the  braces  become  ties,  and  the  ties 
braces,  possessing  the  same  strength  in  both  positions. 

The  foregoing  remarks  will,  it  is  believed,  enable  the  reader 
to  understand  the  objects  of  the  proposed  improvements  and 
the  principles  on  which  they  are  founded. 

1st.  The  braces,  instead  of  being  single,  as  in  the  common 
lattice,  are  in  pairs,  one  on  each  side  of  the  truss,  between 
which  a  vertical  tie  passes ;  this  arrangement  increases  the 
stiffness  upon  the  same  principle  that  a  hollow  cylinder  is 
more  stiff  than  a  solid  one  with  the  same  quantity  of  material, 
and  of  the  same  length,  and  obviates  the  defect  of  warping. 


264  BRIDGE   CONSTRUCTION. 

2d.  The  tie  is  vertical,  or  perpendicular  to  the  lower  ehorc^ 
a  position  which  is  more  natural,  and  in  which  it  is  more  effi 
cacious  than  when  inclined. 

3d.  The  end-braces  all  rest  on  and  radiate  from  the  abut 
ment,  by  which  means  a  firm  support  is  given  to  the  structure, 
and  the  truss  is  not  required  of  greater  length  than  is  sufficient 
to  give  the  braces  room. 

4th.  The  truss  is  effectually  counter-braced,  the  braces  be 
coming  ties,  and  the  ties  braces,  when  called  into  action  by  a 
variable  load,  and  are  capable  of  opposing  a  resistance  on  the 
principle  of  the  inclined  tie  of  the  ordinary  lattice  bridge. 

It  is  readily  admitted  that  the  strength  in  the  inverted  is 
less  than  in  the  erect  position,  but  it  must  be  remembered  that 
the  unloaded  bridge  is  always  in  equilibrium ;  that  the  action 
of  the  parts  which  renders  counter-bracing  necessary,  results 
entirely  from  the  variable  load,  and  that,  therefore,  a  combina 
tion  of  timbers  to  resist  its  effects  should  not  be  as  strong  as 
that  which  sustains  both  the  permanent  and  the  variable  loads. 

Behind  the  ends  of  the  lower  chords  at  the  abutments,  and 
between  them  over  the  piers,  double  wedges  are  driven,  the 
object  of  which  is,  by  the  compression  which  they  produce,  to 
relieve  the  tension  of  the  lower  chord. 

For  ordinary  spans,  the  dimensions  of  the  timbers  may  be : 
Braces  2  in.  by  10  in.  in  pairs. 

Ties  3     "      12  " 

Arches  or  arch-braces  6     "       12  " 
Chords  3     "       14  "  lapped. 

Pins  *2J  in.  in  diameter. 

In  conclusion,  it  is  proper  to  remark  that  the  proposed  plan 
is  not  recommended  as  the  best  under  all  circumstances,  but 
it  is  as  economical  in  first  cost  as  any  other  that  can  be  used. 
The  arrangement  will  be  found  even  more  simple  than  the 
ordinary  lattice,  and  it  is  equally  applicable  for  bridges  on 
common  roads  or  railroads,  and  for  roof  or  deck  bridges.  The 
braces,  in  consequence  of  being  placed  in  pairs,  require  a  slight 
increase  of  timber  over  the  common  plan,  in  the  proportion  of 
40  to  36,  but  the  diminished  lengths  of  the  ties  and  of  the  truss 
more  than  counterbalance  this  increase. 


TRUSSED   GIRDER    BRIDGES.  265 

The  cost  of  workmanship  on  the  truss  is  very  trifling,  and 
less  than  on  the  common  lattice ;  if  the  timbers  are  cut  to  the 
proper  lengths,  the  auger  will  be  the  only  tool  required  in  put 
ting  it  together. 


TRUSSED  GIRDER  BRIDGES.     (Plate  15.) 

Much  diversity  of  opinion  exists  in  regard  to  the  manner 
of  constructing  trussed  girder  bridges,  and  the  true  relative 
proportion  of  the  beams  and  tension  rods.  It  is  asserted  by 
some  that  small  tension  rods  are  worse  than  useless,  others 
think  that  even  the  slightest  rod  must  render  some  assistance, 
and  that  the  strength  will  be  increased  by  any  addition  of  this 
kind.  A  practical  illustration  of  the  subject  can  best  be  given 
by  an  example,  and  a  calculation  will  be  made  from  the  follow 
ing  data. 

Span  between  supports  50  feet. 

Between  points  of  attachment  of  rods  54    " 

Middle  interval  14    " 

End  intervals  18     " 

Number  of  girders  2 

Distance  of  middle  of  rod  below  middle  of  girder     3  feet. 
The  first  hypothesis  will  be  that  the  truss  rods  are  two  in 
number,  each  1  inch  in  diameter. 

The  span  being  50  feet,  and  the  distance  of  the  rod  below 
middle  of  girder  3  feet,  the  length  of  rod  between  edges  of 
abutments  will  be  14  +  2  V182  +  32  =  50,496  feet. 

The  deflection  of  the  beam  itself,  allowing  the  load  to  be 
L  ton  per  foot  uniformly  distributed,  and  the  weight  of  timber 
3  pounds  per  foot  B.  M.,  will  be  determined  from  the  expres 
sion  10  —  — T^ — -  in  which  the  deflection  is  supposed  to  be  ?*» 

of  an  inch  for  1  foot  in  length,  or  f  g  =  1-J  inches  in  50  feet ;  the 
weight  being  applied  at  centre.     By  substituting  the  proper 

80  x  20  x  203 
frames,  we  nave  w  — ^ —    -  =  5,120  pounds. 


266  BRIDGE   CONSTItCTCTION. 

The  deflection  of  the  beam,  therefore,  on  the  supposition 
that  it  does  not  break,  but  preserves  its  elasticity,  will  be 
5,120  :  55,000:  :  1-1-  :  13J  =  total  deflection  in  inches  on  thig 
hypothesis. 

But  the  actual  deflection  of  the  trussed  beam  will  not  be  as 
great  as  this,  since  the  rods  will  resist  a  portion  of  the  strain, 
and  consequently  the  beam  will  be  to  some  extent  relieved  as 
long  as  the  rods  remain  unbroken. 

The  result  of  experiments  on  the  strength  of  wrought-iron 
is,  that  in  perfect  specimens  the  elasticity  is  not  impaired,  and 
consequently  the  strength  is  not  injured  by  a  weight  which 
does  not  exceed  15,000  pounds  per  square  inch.  Of  course  a 
weight  greater  than  the  elastic  limit  must  eventually  cause  the 
fracture  of  the  material ;  when  it  begins  to  lose  its  elasticity ', 
it  begins  to  break. 

Assuming  the  resistance  therefore  at  15,000  pounds  per 
square  inch  acting  as  a  force  of  tension  on  the  rods,  the  weight 

which  it  would  sustain  at /or  g'y  will  be  — TcToIa —  ~  2,470 

pounds,  equivalent  to  a  uniform  load  over  the  whole  beam,  of 
IT  :  50  :  :  2,470  :  :  7,265  pounds  for  every  square  inch  in  the 
cross-section  of  the  rods. 

The  actual  cross-section  of  the  rods  being  1-57  square 
inches,  the  actual  weight  that  they  can  sustain  will  be  7,265  x 
1-57  =  11,406  pounds.  Deducting  11,406  pounds  from  55,000 
pounds,  there  remain  43,594  pounds  to  be  sustained  by  the 
beam  itself. 

The  deflection  of  the  beam  by  this  weight  will  be  5,120  : 
43,594  :  :  1J  :  :  10J  inches  =  the  actual  deflection  of  the 
trussed  beam  on  the  supposition  that  the  rod  will  admit  of  a 
sufficient  extension  without  breaking,  or  injury  to  its  elasticity. 
This  point  must  now  be  examined. 

If  the  deflection  becomes  10J  inches  =  -,%  foot,  the  length 
of  the  rod  estimated  as  before,  will  be  14  x  2  N/18^^-3-92^ 
50,836  feet. 

The  extension  will  therefore  be  50/836 —  50,496  =-340  feet 
=  T|T  of  the  length  of  the  rod. 

The  extension  that  iron  is  capable  of  bearing  without  in- 


TRUSSED   GIRDER  BRIDGES.  267 

jury  to  its  elasticity  is  only  y^1^  of  its  length.  Consequently 
the  strain  with  the  dimensions  assumed  will  be  more  than 
nine  times  the  elastic  limit.  Therefore  the  rods  must  break 
before  the  beam  reaches  the  deflection  that  the  weight  must 
necessarily  produce,  and  are  evidently  of  no  assistance. 

The  truth  of  this  conclusion  is  confirmed  by  experience, 
and  practical  men  are  adopting  the  opinion,  based  on  expe 
rience  and  observation,  that  small  rods,  unless  increased  in 
number  to  secure  a  sufficient  resisting  area,  are  of  no  value. 

The  only  correct  and  safe  way  of  proportioning  a  trussed 
girder  bridge  is,  to  assume  the  size  of  the  beams  and  all  other 
dimensions  of  the  structure  except  the  rods,  and  determine  the 
cross-section  of  the  latter  by  the  condition  that  the  strain  per 
square  inch  shall  be  a  given  quantity. 

To  illustrate  this  case,  let  the  same  dimensions  be  continued, 
the  rods  excepted,  and  let  the  maximum  strain  per  square  inch 
be  limited  to  10,000  pounds. 

The  position  of  the  neutral  axis  from  which  the  strains 
should  be  estimated  will  depend, 

1.  On  the  relative  magnitude  of  the  cross-sections  of  the 
rods  and  girders. 

2.  On  the  relative  powers  of  resistance  of  the  material. 

3.  On  the  relative  extensibility  and  compressibility  of  the 
material. 

An  accurate  mathematical  solution  of  the  problem,  although 
not  impossible,  is  nevertheless  too  complicated  for  general  use, 
and  it  is  not  necessary  to  resort  to  it,  as  a  very  near  approxi 
mation,  fully  sufficient  for  all  practical  purposes,  can  be  ob 
tained  without  it. 

As  the  cross-section  of  the  girders,  in  a  bridge  of  the  kind 
under  consideration,  is  always  much  greater  in  proportion  to 
the  extensibility  and  powers  of  resistance  of  the  material  than 
that  of  the  rods,  the  middle  of  the  beam  may  be  assumed  as  a 
fulcrum,  and  the  strain  upon  the  rods  estimated  from  this 
point.  The  uniform  weight  on  the  whole  bridge  being  repre 
sented  by  w,  the  portion  at  the  angle  of  the  tension  rod  will 
be  -5-0  w,  and  the  strain  caused  by  this  weight  will  be  J{j  w 

18-25 
( — Q—  )  in  the  direction  of  the  rod.    As  w  —  110,000  the  strain 


268  BRIDGE    CONSTRUCTION. 

will  be  214,133  pounds,  requiring  (at  10,000  pounds  per  square 
inch)  21  square  inches  in  the  cross-section  of  the  rods,  or  in 
proportion  if  the  strain  should  be  increased  or  diminished. 

This  is  a  greater  proportion  of  iron  than  is  usually  allowed, 
but  it  is  not  too  great  for  security.  The  girders  cannot  oppose 
any  direct  resistance  to  a  cross-strain  without  experiencing 
flexure ;  but  a  railroad  bridge  should  be  as  rigid  as  possible, 
and  therefore  the  rods  should  be  depended  upon  to  resist  the 
whole  of  the  tension,  and  act  as  the  lower  chords  of  an  ordinary 
bridge.  In  this  way  the  calculation  becomes  very  simple,  and 
furnishes  safe  practical  results. 

In  the  construction  of  trussed  girder  bridges,  the  stiffness 
would  be  greatly  increased  by  the  introduction  of  diagonal  ties 
or  braces  in  the  middle  rectangular  interval,  and  with  this  ad 
dition,  and  proportioned  upon  the  principles  above  illustrated, 
it  becomes  a  safe,  economical,  and  in  every  respect  a  good 
Bridge  for  moderate  spans. 


THE   END. 


THIS  SiraQUEHANNA 


CROSS  SECTION. 


PLAN  OF  TOP  CHORD. 


••PLAN  Oh'  LOVVKR. 


ELKVATIflM  OD1  f^OWER.  rJHORT).. 


Plate 


R12ABUTC  M.  B. 

ARCHED  TRUSS  BETDG-E. 


POST  CHORD  ARCH  FASTENING. 


COVE  HUN  VIADUCT 


.'Plate  3 


SECTION  OF  AR.CE. 


"d 


BALTIMORE  AsSUSqiTBH 

TUBULAR  PLATE  IRON"  BRIDGE 


ULiSi. 


3 


PLAN. 


M  I  I  I  I  I  J  Q 


GROSS    SECT LOW 


TBMIIOXG'E  OVE 

Details 
TOP  CHORD.  BOTTOM  OHOBD 


h     h 
~ 


mor. 


BEARING  BLOCK 

S^f 

A^ £kj 


Plati 


Pk,te  6. 


LITTLE  JUNIATA  J3E.IDG-E. 


SECT  ION  OP  TOP  CHORDS. 


BOTTOM/  |  CHORD. 

\ 


SECTION  OF  BOTTOM 
CHORDS. 


\\  o  / 

-6- 


-Y- 


TOP    CHOBD. 


SKEW  BACK'. 


BRID&E   UM  PJEJfSr*  M.  B.. 

DYER 

SHERMAN'S    CREEK 


GE.OSS  SECTlOiV 


CHORDS. 


VIEW  OF 


Plat 


flXDKB.18  "SPAT  JEST  IJRflMST  BRIDGE* 


fett 


X 


: 


CBOSS  SEGTiOK 


UPPER   CHOED 


lOWER, 


CH  IB 


,,s> 


c^ 


LOWEU  CHOR11. 


Plate 


CRUSLS  SECTION 


PORTION   OF  TflUSS 


e   10 


Kjrar*  m.  IB.. 

CANAL  BRIDGE. 


CROSS  (SECTION. 


ELEVATION  01-'  AtUlll. 


SECTION. 


SET  [ft]  SHREW". 


SlGCTION  OF  I.nV/l'.R    CIHORP   &r  A.RHH. 


Tlate  11 


IRON   BHIDGE   AT  RACOON  GREEK. 


SECTION.  TOP    CHORDS.  BOTTOM 

1 


^J  E 


y 


1 


PORTION  OP  AB.GE.  SOCKET  BOX. SMALL 


PEDESTAI.. 


m 


o 


GAE  PLATE 

, 

LARGE  CYLINDER. 


SKEWBAGK. 


LJ 

i 

c 

c 

r 

-   —  k 

OHIO  11. 1L 

IMPROVED  ARCH   BRACE    TRUSS    FRAME. 


Fig.   1. 


.  4 . 


Hate  13 


AT   Gi  AUK'S  FERRY". 


CROSS 


TOWING  PAT 


PIjATST  OF  ATA.GH  AND  POST 


14. 


IiATTICB   TRUSS. 


SECTION"  THROUGH  POST. 


SECTION"  TH-ROUG-H  TI]iI 


GIKDKR  BKIUGE  . 


50  REET  SPAN 


i W 


\\ 


CR0 


UNDER  VIEW  OF  TENSI01I  "ROJ3  S- 


:  1  A,B. 


w 


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i  i 

N  3 


* 


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